c 2 ˘3/ 4 @ 4 . 8 d b =/ ˘ ˜?5˜#1/ - uniba.sk · been generalized for more complex option...

Department of Applied Mathematics and Statistics http://www.iam.fmph.uniba.sk/institute/sevcovic/knihy/index.html

2 of 2 01/23/09 13:37

Daniel Ševčovič: Transformation methods for evaluating approximations to the optimal

exercise boundary for linear and nonlinear Black-Scholes equations. In: M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance:New Research Trends in Option Pricing, pp. 153-198, 2008 Nova Science Publishers, Inc., Hauppauge ISBN 978-1-60456-931-5 Download chapter

Department of Applied Mathematics and Statistics, Divison of Applied Mathematics

Faculty of Mathematics, Physics and Informatics Comenius University

Mlynska dolina 842 48 Bratislava, Slovak republic

www.iam.fmph.uniba.sk/institute/sevcovic

e-mail:[email protected]

Tel: + 421-2-654 24 000 Fax:+ 421-2-654 25 882

arX

iv:0

805.

0611

v1 [

mat

h.A

P]

5 M

ay 2

008

Transformation methods for evaluating approximationsto the optimal exercise boundary for linear and nonlinear

Black-Scholes equations

DanielSevcovic1

1 Department of Applied Mathematics and Statistics,Faculty of Mathematics, Physics & Informatics, Comenius University,

842 48 Bratislava, Slovak Republic,[email protected]

Abstract. The purpose of this survey chapter is to present a transformation technique that canbe used in analysis and numerical computation of the early exercise boundary for an American styleof vanilla options that can be modelled by class of generalized Black-Scholes equations. We an-alyze qualitatively and quantitatively the early exerciseboundary for a linear as well as a class ofnonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear functionof the second derivative of the option price itself. A motivation for studying the nonlinear Black-Scholes equation with a nonlinear volatility arises from option pricing models taking into accounte.g. nontrivial transaction costs, investor’s preferences, feedback and illiquid markets effects andrisk from a volatile (unprotected) portfolio. We present a method how to transform the free bound-ary problem for the early exercise boundary position into a solution of a time depending nonlinearnonlocal parabolic equation defined on a fixed domain. We furthermore propose an iterative numer-ical scheme that can be used in order to find an approximation of the free boundary. In the case ofa linear Black-Scholes equation we are able to derive a nonlinear integral equation for the positionof the free boundary. We present results of numerical approximation of the early exercise boundaryfor various types of linear and nonlinear Black-Scholes equations and we discuss dependence of thefree boundary on model parameters. Finally, we discuss an application of the transformation methodfor the pricing equation for American type of Asian options.

Contents

1 Introduction 2

2 Risk adjusted methodology model 72.1 Risk adjusted Black–Scholes equation . . . . . . . . . . . . . . .. . . . . . . . . 92.2 Pricing of European style of options by the RAPM model . . .. . . . . . . . . . . 11

3 Transformation method for a linear Black–Scholes equation 133.1 Fixed domain transformation of the free boundary problem . . . . . . . . . . . . . 153.2 Reduction to a nonlinear integral equation . . . . . . . . . . .. . . . . . . . . . . 173.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 203.4 Early exercise boundary for an American put option . . . . .. . . . . . . . . . . . 24

1

http://arXiv.org/abs/0805.0611v1

2 DanielSevcovic

4 Transformation method for a nonlinear Black–Scholes equation 254.1 An iterative algorithm for approximation of the early exercise boundary . . . . . . 284.2 Numerical approximations of the early exercise boundary . . . . . . . . . . . . . . 31

4.2.1 Case of a constant volatility – comparison study . . . . .. . . . . . . . . . 314.2.2 Risk Adjusted Pricing Methodology model . . . . . . . . . . .. . . . . . 334.2.3 Barles and Soner model . . . . . . . . . . . . . . . . . . . . . . . . . . .36

5 Transformation methods for Asian call options 365.1 Governing equations for Asian options . . . . . . . . . . . . . . .. . . . . . . . . 375.2 American style of Asian options . . . . . . . . . . . . . . . . . . . . .. . . . . . 385.3 Fixed domain transformation for American style of Asiancall options . . . . . . . 395.4 An approximation scheme for pricing American style of Asian options . . . . . . . 415.5 Computational examples of the free boundary approximation . . . . . . . . . . . . 43

1 Introduction

According to the classical theory due to Black, Scholes and Merton the price of an optionin an idealized financial market can be computed from a solution to the well-known Black–Scholes linear parabolic equation derived by Black and Scholes in [7], and, independentlyby Merton (see also Kwok [38], Dewynneet al. [16], Hull [30], Wilmott et al. [53]). Recallthat a European call (put) option is the right but not obligation to purchase (sell) an under-lying asset at the expiration priceE at the expiration timeT . Assuming that the underlyingassetS follows a geometric Brownian motion

dS = ( − q)Sdt + σSdW, (1)

where is a drift, q is the asset dividend yield rate,σ is the volatility of the asset andWis the standard Wiener process (cf. [38]), one can derive a governing partial differentialequation for the price of an option. We remind ourselves thatthe equation for option’s priceV (S, t) is the following parabolic PDE:

∂V

∂t+ (r − q)S

∂V

∂S+

σ2

2S2 ∂2V

∂S2− rV = 0 (2)

whereσ is the volatility of the underlying asset price process,r > 0 is the interest rate of azero-coupon bond,q ≥ 0 is the dividend yield rate. A solutionV = V (S, t) represents theprice of an option if the price of an underlying asset isS > 0 at timet ∈ [0, T ].

The case when the diffusion coefficientσ > 0 in (2) is constant represents a classicalBlack–Scholes equation originally derived by Black and Scholes in [7]. On the other hand,if we assume the volatility coefficientσ > 0 to be a function of the solutionV itself then (2)with such a diffusion coefficient represents a nonlinear generalization of the Black–Scholesequation. It is a purpose of this chapter to focus our attention to the case when the diffusioncoefficientσ2 may depend on the timeT − t to expiry, the asset priceS and the secondderivative∂2

SV of the option price (hereafter referred to asΓ), i.e.

σ = σ(S2∂2SV, S, T − t) . (3)

Transformation methods for linear and nonlinear Black-Scholes equations 3

A motivation for studying the nonlinear Black–Scholes equation (2) with a volatility σhaving a general form (3) arises from option pricing models taking into account nontrivialtransaction costs, market feedbacks and/or risk from a volatile (unprotected) portfolio. Re-call that the linear Black–Scholes equation with constantσ has been derived under severalrestrictive assumptions like e.g. frictionless, liquid and complete markets, etc. We alsorecall that the linear Black–Scholes equation provides a perfectly replicated hedging port-folio. In the last decades some of these assumptions have been relaxed in order to model,for instance, the presence of transaction costs (see e.g. Leland [39], Hoggardet al. [29],Avellaneda and Paras [4]), feedback and illiquid market effects due to large traders choos-ing given stock-trading strategies (Frey and Patie [22], Frey and Stremme [23], Duringet al. [17], Schonbucher and Wilmott [49]), imperfect replication and investor’s prefer-ences (Barles and Soner [8]), risk from unprotected portfolio (Kratka [37], Jandacka andSevcovic [32] or [47]). One of the first nonlinear models isthe so-called Leland model(cf. [39]) for pricing call and put options under the presence of transaction costs. It hasbeen generalized for more complex option strategies by Hoggard, Whaley and Wilmottin [29]. In this model the volatilityσ is given by

σ2(S2∂2SV, S, τ) = σ2(1 + Le sgn(∂2

SV )) (4)

whereσ > 0 is a constant historical volatility of the underlying assetprice process andLe > 0 is the so-called Leland constant given by Le=

√

2/πC/(σ√

∆t) whereC > 0 isa constant round trip transaction cost per unit dollar of transaction in the assets market and∆t > 0 is the time-lag between portfolio adjustments.

Notice that dependence of volatility adjustment on the second derivative of the priceis quite natural. Indeed, in the idealized Black-Scholes theory, the optimal hedge is equalto ±∂SV and therefore one may expect more frequent transaction in regions with the highsecond derivative∂2

SV (cf. [7]).A popular nonlinear generalization of the Black–Scholes equation has been proposed

by Avellaneda, Levy, and Paras [5] for description of incomplete markets and uncertain butbounded volatility. In their model we have

σ2(S2∂2SV, S, τ) =

{

σ21 if ∂2

SV < 0,σ2

2 if ∂2SV > 0,

(5)

whereσ1 andσ2 represent a lower and upper a-priori bound on the otherwise unspecifiedasset price volatility.

If transaction costs are taken into account perfect replication of the contingent claimis no longer possible and further restrictions are needed inthe model. By assuming thatinvestor’s preferences are characterized by an exponential utility function Barles and Soner(cf. [8]) derived a nonlinear Black–Scholes equation with the volatility σ given by

σ2(S2∂2SV, S, τ) = σ2

(

1 + Ψ(a2erτS2∂2SV )

)

(6)

whereΨ is a solution to the ODE:Ψ′(x) = (Ψ(x) + 1)/(2√

xΨ(x) − x),Ψ(0) = 0, and

a > 0 is a given constant representing risk aversion. Notice thatΨ(x) = O(x13 ) for x → 0

andΨ(x) = O(x) for x → ∞.

4 DanielSevcovic

Another popular model has been derived for the case when the asset dynamics takes intoaccount the presence of feedback and illiquid market effects. Frey and Stremme (cf. [22,23]) introduced directly the asset price dynamics in the case when a large trader chooses agiven stock-trading strategy (see also [49]). The diffusion coefficientσ is again nonconstantand it can be expressed as:

σ2(S2∂2SV, S, τ) = σ2

(

1 − λ(S)S∂2SV)−2

(7)

whereσ2, > 0 are constants andλ(S) is a strictly convex function,λ(S) ≥ 1. Interest-ingly enough, explicit solutions to the Black–Scholes equation with varying volatility as in(7) have been derived by Bordag and Chankova [9] and Bordag and Frey [10].

The last example of the Black–Scholes equation with a nonlinearly depending volatilityis the so-called Risk Adjusted Pricing Methodology model proposed by Kratka in [37] andrevisited by Jandacka andSevcovic in [32]. In order to maintain (imperfect) replication ofa portfolio by the delta hedge one has to make frequent portfolio adjustments leading to asubstantial increase in transaction costs. On the other hand, rare portfolio adjustments maylead to an increase of the risk arising from a volatile (unprotected) portfolio. In the RAPMmodel the aim is to optimize the time-lag∆t between consecutive portfolio adjustments.By choosing∆t > 0 in such way that the sum of the rate of transaction costs and the rate ofa risk from unprotected portfolio is minimal one can find the optimal time lag∆t > 0. Inthe RAPM model, it turns out that the volatility is again nonconstant and it has the followingform:

σ2(S2∂2SV, S, τ) = σ2

(

1 + µ(S∂2SV )

13

)

. (8)

Here σ2 > 0 is a constant historical volatility of the asset price returns andµ =

3(C2R/2π)13 whereC,R ≥ 0 are nonnegative constant representing the transaction cost

measure and the risk premium measure, resp. (see [32] for details).Notice that all the above mentioned nonlinear models are consistent with the original

Black–Scholes equation in the case the additional model parameters (e.g. Le,a, , µ) arevanishing. If plain call or put vanilla options are concerned then the functionV (S, t) isconvex inS variable and therefore each of the above mentioned models has a diffusioncoefficient strictly larger thanσ2 leading to a larger values of computed option prices. Theycan be therefore identified with higher Ask option prices, i.e. offers to sell an option.Furthermore, these models have been considered and analyzed mostly for European styleof options, i.e. options that can be exercised only at the maturity t = T . On the other hand,American options are much more common in financial markets asthey allow for exercisingof an option anytime before the expiryT . In the case of an American call option a solutionto equation (2) is defined on a time dependent domain0 < S < Sf (t), 0 < t < T . It issubject to the boundary conditions

V (0, t) = 0 , V (Sf (t), t) = Sf (t) − E , ∂SV (Sf (t), t) = 1 , (9)

and terminal pay-off condition at expiryt = T

V (S, T ) = max(S − E, 0) (10)


whereE > 0 is a strike price (cf. [16, 38]). One of important problems inthis field isthe analysis of the early exercise boundarySf (t) and the optimal stopping time (an inversefunction toSf (t)) for American call (or put) options on stocks paying a continuous dividendyield with a rateq > 0 (or q ≥ 0). However, an exact analytical expression for the freeboundary profile is not even known for the case when the volatility σ is constant. Manyauthors have investigated various approximation models leading to approximate expressionsfor valuing American call and put options: analytic approximations (Barone–Adesi andWhaley [6], Kuske and Keller [36], Dewynneet al. [16], Geskeet al. [24, 25], MacMillan[40], Mynemi [44]); methods of reduction to a nonlinear integral equation (Alobaidi [1],Kwok [38], Mallier et al. [41, 42], Sevcovic [46], Stamicaret al. [50]). In recent papers[55, 56], Zhu derived a closed form of the free boundary position in terms of on infiniteseries. We also refer to a recent survey paper by Chadam [12] focusing on free boundaryproblems in mathematical finance.

We remind ourselves that in the case of a constant volatilitythere are, in principle, twoways how to solve numerically the free boundary problem for the value of an American callresp. put option and the position of the early exercise boundary. The first class of algorithmsis based on reformulation of the problem in terms of a variational inequality (see Kwok [38]and references therein). The variational inequality can bethen solved numerically by theso-called Projected Super Over Relaxation method (PSOR forshort). An advantage of thismethod is that it gives us immediately the value of a solution. A disadvantage is that onehas to solve large systems of linear equations iteratively taking into account the obstacle fora solution, and, secondly, the free boundary position should be deduced from the solution aposteriori. Moreover, the PSOR method is not directly applicable for solving the problem(2)-(9) when the diffusion coefficientσ may depend on the second derivative of a solutionitself. The second class of methods is based on derivation ofa nonlinear integral equationfor the position of the free boundary without the need of knowing the option price itself (seee.g. Evans, Kuske-Keller [21,36], Mallier and Alobaidi [1,41,42],Sevcovicet al. [46,50],Chadamet al. [12, 13]. In this approach an advantage is that only a single equation for thefree boundary has to be solved provided thatσ is constant; a disadvantage is that the methodis based on integral transformation techniques and therefore the assumptionσ is constant iscrucial.

Higher order finite difference approximations of the free boundary problems for callor put options are discussed in recent papers by Ankudinova and Ehrhardt [2, 3], Ehrhardtand Mickens [18] and Zhaoet al. [54]. Other interesting analytical and numerical methodsfor evaluating the early exercise boundary have been recently studied by e.g. Milsteinet al. [43] (Monte–Carlo methods), Widdickset al. [52] (singular pertubation techniquesand asymptotic expansions), Choet al. [14] (parameter estimation methods), Grandits andSchachingeret al. [26] (tracking of a discontinuity method), Imaiet al. [31] (numericalmethod for generalized Lelend model).

In this survey we recall an iterative numerical algorithm for solving the free boundaryproblem for an American type of options in the case the volatility σ may depend on theoption and asset values as well as on the timeT − t to expiry as well as for American type

6 DanielSevcovic

of Asian options. A key idea of this method consists in transformation of the free boundaryproblem into a semilinear parabolic equation defined on a fixed spatial domain coupledwith a nonlocal algebraic constraint equation for the free boundary position. It has beenproposed and analyzes by the author in a series of papers [46–48, 50]. Since the resultingparabolic equation contains a strong convective term we make use of the operator splittingmethod in order to overcome numerical difficulties. A full space-time discretization ofthe problem leads to a system of semi-linear algebraic equations that can be solved by aniterative procedure at each time level.

The rest of the chapter has the following organization: in the next section we recall thewell known nonlinear generalization of Black–Scholes equations due to Frey and Stremme,Barles and Soner and Kratka, Jandacka andSevcovic, resp. We also present qualitativeand quantitative properties of the nonlinear models for pricing European style of optionswith special focus on the Risk adjusted pricing methodology(RAPM) due to Kratka [37],Jandacka andSevcovic [32]. Section 3 is devoted to the free boundary problem for pricingAmerican options by means of a linear Black–Scholes equation, i.e. σ > 0 is constant.This is important in order to understand important steps of the fixed domain transformationmethod. A resulting system of transformed equations consists of a nonlocal parabolic equa-tion defined on a fixed domain with time depending coefficientsand an algebraic constraintequation for the free boundary position. Since the volatility σ is constant in this case, byusing Sine and Cosine Fourier transformations the system oftransformed equations can befurther simplified and reduced to a single nonlinear integral equation for the free boundaryposition. We show how this integral equation can be utilizedin order to obtain qualitativeproperties of the free boundary position (early exercise behavior, long time behavior) aswell as quantitative properties (a fast and stable numerical scheme for computing the freeboundary function). In Section 4 we discuss a transformation method applied to a class ofnonlinear Black–Scholes equations. We are able to derive a similar system of transformedparabolic–algebraic equation to the one from Section 3. However, as the volatility is nolonger a constant and it may depend on the solution itself theresulting system of equa-tions can not be reduced to a single integral equation for thefree boundary and it has tobe solved numerically. We propose a numerical method based on the finite difference ap-proximation combined with an operator splitting techniquefor numerical approximation ofthe solution and computation of the free boundary position.Several numerical results fornonlinear Black–Scholes equations with volatility functionsσ defined as in (6) and (8) arepresented. We also compare our methodology with well-knownmethods for evaluation ofapproximation to the free boundary position in the case the volatility σ is constant. We an-alyze dependence of the free boundary position with respectto various parameters enteringexpressions (6) and (8). Finally, Section 5 is devoted to a recent application of the trans-formation method in the case of American style of Asian options in which the strike priceis an arithmetical average of underlying asset prices. Although the volatilityσ is assumedto be constant, due to a specific character of the Black–Scholes equation for pricing Asianoption the resulting transformed system of equation cannotbe further reduced and has to besolved numerically by a slight modification of the numericalmethod discussed in Section


4. We finish the last section by presentation of several illustrative examples describing theearly exercise boundary for American style of Asian call options with floating strike.

2 Risk adjusted methodology model

The aim of this section is to present one of nonlinear generalizations of the classical Black–Scholes equation with a volatilityσ of the form (3) in a more detail. We focus on theso-called Risk adjusted pricing methodology model due to Kratka [37] and straightforwardgeneralization by Jandacka andSevcovic [32] (see also [47]). In this model both the riskarising from nontrivial transaction costs as well as the risk from unprotected volatile port-folio are taken into account. Their sum representing the total risk is subject of minimiza-tion. The original model was proposed by [37]. In [32] we modified Kratka’s approachby considering a different measure for risk arising from unprotected portfolio in order toconstruct a model which is scale invariant and mathematically well posed. These two im-portant features were missing in the original model of Kratka. The model is based on theBlack–Scholes parabolic PDE in which transaction costs aredescribed by the Hoggard,Whalley and Wilmott extension of the Leland model (cf. [29,30,38]) whereas the risk froma volatile portfolio is described by the average value of thevariance of the synthesizedportfolio. Transaction costs as well as the volatile portfolio risk depend on the time-lagbetween two consecutive transactions. We define the total risk premium as a sum of trans-action costs and the risk cost from the unprotected volatileportfolio. By minimizing thetotal risk premium functional we obtain the optimal length of the hedge interval.

Concerning the dynamics of an underlying asset we will assume that the asset priceS = S(t), t ≥ 0, follows a geometric Brownian motion (1) with a driftρ, standard deviationσ > 0 and it may pay continuous dividends, i.e.dS = (ρ − q)Sdt + σSdW wheredWdenotes the differential of the standard Wiener process andq ≥ 0 is a continuous dividendyield rate. This assumption is usually made when deriving the classical Black–Scholesequation (see e.g. [30, 38]). Similarly as in the derivationof the classical Black–Scholesequation we construct a synthesized portfolioΠ consisting of a one option with a priceVandδ assets with a priceS per one asset:

Π = V + δS . (11)

We recall that the key idea in the Black–Scholes theory is to examine the differential∆Πof equation (11). The right-hand side of (11) can be differentiated by using Ito’s formulawhereas portfolio’s increment∆Π(t) = Π(t + ∆t) − Π(t) of the left-hand side can beexpressed as follows:

∆Π = rΠ∆t + δqS∆t (12)

wherer > 0 is a risk-free interest rate of a zero-coupon bond. In the real world, such asimplified assumption is not satisfied and a new term measuring the total risk should beadded to (12). More precisely, the change of the portfolioΠ is composed of two parts:the risk-free interest rate partrΠ∆t and the total risk premium:rRS∆t whererR is a risk

8 DanielSevcovic

premium per unit asset price. We consider a short positionedcall option. Therefore thewriter of an option is exposed to this total risk. Hence we aregoing to price the higher Askoption price – an offer to sell an option. It means that∆Π = rΠ∆t − rRS∆t. The totalrisk premiumrR consists of the transaction risk premiumrTC and the portfolio volatilityrisk premiumrV P , i.e. rR = rTC + rV P . Hence

∆Π = rΠ∆t + δqS∆t − (rTC + rV P )S∆t . (13)

Our next goal is to show how these risk premium measuresrTC , rV P depend on the time lagand other quantities, like e.g.σ, S, V, and derivatives ofV. The problem can be decomposedin two parts: modeling the transaction costs measurerTC and volatile portfolio risk measurerV P .

We begin with modeling transaction costs. In practice, we have to adjust our portfolioby frequent buying and selling of assets. In the presence of nontrivial transaction costs,continuous portfolio adjustments may lead to infinite totaltransaction costs. A natural wayhow to consider transaction costs within the frame of the Black–Scholes theory is to followthe well known Leland approach extended by Hoggard, Whalleyand Wilmott (cf. [29,38]).We will recall an idea how to incorporate the effect of transaction costs into the governingequation. More precisely, we will derive the coefficient of transaction costsrTC occurringin (13). Let us denote byC the round trip transaction cost per unit dollar of transaction.Then

C = (Sask − Sbid)/S (14)

whereSask andSbid are the so-called Ask and Bid prices of the asset, i.e. the market priceoffers for selling and buying assets, resp. HereS = (Sask + Sbid)/2 denotes the mid valueof the underlying asset price.

In order to derive the termrTC in (13) measuring transaction costs we will assume,for a moment, that there is no risk from volatile portfolio, i.e. rV P = 0. Then∆V +δ∆S = ∆Π = rΠ∆t + δqS∆t + rTCS∆t. Following Leland’s approach (cf. [29]), usingIto’s formula and assumingδ-hedging of a synthetised portfolioΠ one can derive that thecoefficientrTC of transaction costs is given by the formula:

rTC =CσS√

2π

∣

∣∂2SV∣

∣

1√∆t

(15)

(see [29, Eq. (3)]). It leads to the well known Leland generalization of the Black–Scholesequation (2) in which the diffusion coefficient is given by (4) (see [29,39] for details).

Next we focus our attention to the problem how to incorporatea risk from a volatileportfolio into the model. In the case when a portfolio consisting of options and assets ishighly volatile, an investor usually asks for a price compensation. Notice that exposure torisk is higher when the time-lag between portfolio adjustments is higher. We shall propose ameasure of such a risk based on the volatility of a fluctuatingportfolio. It can be measuredby the variance of relative increments of the replicating portfolio Π = V + δS, i.e. bythe termvar((∆Π)/S). Hence it is reasonable to define the measurerV P of the portfolio


volatility risk as follows:

rV P = Rvar

(

∆ΠS

)

∆t. (16)

In other words,rV P is proportional to the variance of a relative change of a portfolio pertime interval∆t. A constantR represents the so-calledrisk premium coefficient. It canbe interpreted as the marginal value of investor’s exposureto a risk. If we apply Ito’sformula to the differential∆Π = ∆V + δ∆S we obtain∆Π = (∂SV + δ) σS∆W +12σ2S2Γ(∆W )2 +G whereΓ = ∂2

SV andG = (∂SV + δ)ρS∆t+∂tV ∆t is a deterministicterm, i.eE(G) = G in the lowest order∆t - term approximation. Thus

∆Π − E(∆Π) = (∂SV + δ) σSφ√

∆t +1

2σ2S2(φ2 − 1)Γ∆t

whereφ is a random variable with the standard normal distribution such that∆W = φ√

∆t.Hence the variance of∆Π can be computed as follows:

var(∆Π) = E(

[∆Π − E(∆Π)]2)

= E(

[(∂SV + δ)σSφ√

∆t + 12σ2S2Γ

(

φ2 − 1)

∆t]2)

.

Similarly, as in the derivation of the transaction costs measurerTC we assume theδ-hedgingof portfolio adjustments, i.e. we chooseδ = −∂SV . SinceE((φ2 − 1)2) = 2 we obtain anexpression for the risk premiumrV P in the form:

rV P =1

2Rσ4S2Γ2∆t . (17)

Notice that in our approach the increase in the time-lag∆t between consecutive transac-tions leads to a linear increase of the risk from a volatile portfolio where the coefficientof proportionality depends the asset priceS, option’s Gamma,Γ = ∂2

SV , as well as theconstant historical volatilityσ and the risk premium coefficientR.

2.1 Risk adjusted Black–Scholes equation

The total risk premiumrR = rTC + rV P consists of two parts: transaction costs premiumrTC and the risk from a volatile portfoliorV P premium defined as in (15) and (17), resp.We assume that an investor is risk aversive and he/she wants to minimize the value of thetotal risk premiumrR. For this purpose one has to choose the optimal time-lag∆t betweentwo consecutive portfolio adjustments. As bothrTC as well asrV P depend on the time-lag∆t so does the total risk premiumrR. In order to find the optimal value of∆t we have tominimize the following function:

∆t 7→ rR = rTC + rV P =C|Γ|σS√

2π

1√∆t

+1

2Rσ4S2Γ2∆t .

The unique minimum of the function∆t 7→ rR(∆t) is attained at the time-lag∆topt =

K2/(σ2|SΓ| 23 ) whereK = (C/(R√

2π)13 . For the minimal value of the function∆t 7→

10 DanielSevcovic

Figure 1:The total risk premiumrR = rTC + rV P as a function of the time-lag∆t between twoconsecutive portfolio adjustments.

rR(∆t) we have

rR(∆topt) =3

2

(

C2R

2π

)

13

σ2|SΓ| 43 . (18)

Taking into account both transaction costs as well as risk from a volatile portfolio effectswe have shown that the equation for the change∆Π of a portfolioΠ read as:

∆V + δ∆S = ∆Π∆t = rΠ∆t + δqS∆t − rRS∆t

whererR represents the total risk premium,rR = rTC + rV P . Applying Ito’s lemmato a smooth functionV = V (S, t) and assuming theδ-hedging strategy for the portfolioadjustments we finally obtain the following generalizationof the Black–Scholes equationfor valuing options:

∂V

∂t+

σ2

2S2 ∂2V

∂S2+ (r − q)S

∂V

∂S− rV − rRS = 0 .

By taking the optimal value of the total risk coefficientrR derived as in (18) the option priceV is a solution to the following nonlinear parabolic equation:

(Risk adjusted Black–Scholes equation)

∂V

∂t+

σ2

2S2(

1 + µ(S∂2SV )

13

) ∂2V

∂S2+ (r − q)S

∂V

∂S− rV = 0 , (19)

whereµ = 3(

C2R2π

)13

andup with u = S∂2SV andp = 1/3 stands for the signed power

function, i.e. up = |u|p−1u. In the case there are neither transaction costs (C = 0) northe risk from a volatile portfolio (R = 0) we haveµ = 0. Then equation (19) reducesto the original Black–Scholes linear parabolic equation (2). We note that equation (19)is a backward parabolic PDE if and only if the functionβ(H) = σ2

2 (1 + µH13 )H is an

increasing function in the variableH := SΓ = S∂2SV . It is clearly satisfied ifµ ≥ 0 and

H ≥ 0.


As it is usual in the classical Black-Scholes theory for European style of options (cf.[30,38]) we consider the change of independent variables:x := ln

(

SE

)

, x ∈ R , τ :=T − t , τ ∈ (0, T ) . As equation (19) contains the termSΓ = S∂2

SV it is convenient tointroduce the following transformation:H(x, τ) := SΓ = S∂2

SV (S, t). Now we are inposition to derive an equation for the functionH. It turns out that the functionH(x, τ) isa solution to a nonlinear parabolic equation subject to the initial and boundary conditions.More precisely, by taking the second derivative of equation(19) with respect tox we obtain,after some calculations, thatH = H(x, τ) is a solution to the quasilinear parabolic equation

∂H

∂τ=

∂2

∂x2β(H) +

∂

∂xβ(H) + r

∂H

∂x, (20)

τ ∈ (0, T ), x ∈ R (see [32]). Henceforth, we will refer to (20) as aΓ equation. A solutionH to (20) is subjected to the initial condition atτ = 0:

H(x, 0) = H(x) , x ∈ R , (21)

whereH(x) is the Diracδ function H(x) = δ(x). For the purpose of numerical approxi-mation we approximate the initial Dirac delta function byH(x) = N ′(d)/(σ

√τ∗) where

τ∗ > 0 is sufficiently small,N(d) is the cumulative distribution function of the normal dis-tribution, andd =

(

x + (r − q − σ2/2)τ∗/

σ√

τ∗). It corresponds to the valueH = S∂2SV

of a call (put) option valued by a linear Black–Scholes equation with a constant volatilityσ > 0 at the timeT − τ∗ close to expiryT when the time parameter0 < τ∗ ≪ 1 issufficiently small. In the case of call or put options the function H is subjected to boundaryconditions atx = ±∞,

H(−∞, τ) = H(∞, τ) = 0 , τ ∈ (0, T ) . (22)

A numerical discretization scheme of theΓ equation (20) based on finite volume approxi-mation has been discussed in [32] in more details.

2.2 Pricing of European style of options by the RAPM model

Let us denoteV (S, t;C, σ,R) the value of a solution to (19) with parametersC, σ,R. Sup-pose that the coefficient of transaction costsC is known from and is given by (14). In realoption market data we can observe different Bid and Ask prices for an option,Vbid < Vask,resp. Let us denote byVmid the mid value, i.e.Vmid = 1

2(Vbid + Vask). By the RAPM

model we are able to explain such a Bid-Ask spread in option prices. The higher Ask pricecorresponds to a solution to the RAPM model with some nontrivial risk premiumR > 0andC > 0 whereas the mid valueVmid corresponds to a solutionV (S, t) for vanishing riskpremiumR = 0, i.e. to a solution of the linear Black-Scholes equation (2). An illustrativeexample of Bid-Ask spreads captured by the RAPM model is shown in Fig. 2.

To calibrate the RAPM model we seek for a couple(σRAPM , R) of implied RAPMvolatility σRAPM and risk premiumR such thatVask = V (S, t;C, σRAPM , R) andVmid =

12 DanielSevcovic

Figure 2: A comparison of Bid and Ask option prices computed by means ofthe RAPM model.The middle dotted line is the option price computed from the Black-Scholes equation. We choseσ = 0.3, µ = 0.2, r = 0.011, E = 25 andT = 1 (left) andT = 0.3 (right).

Figure 3: Intra-day behavior of Microsoft stocks (April 4, 2003) and shortly expiring call optionswith expiry date April 19, 2003. Computed implied volatilities σRAPM and risk premium coeffi-cientsR.

V (S, t;C, σRAPM , 0). Such a system of nonlinear equation forσRAPM and R can besolved by by means of the Newton-Kantorovich iterative method (cf. [32]).

As an example we considered sample data sets for call optionson Microsoft stocks. Weconsidered a flat interest rater = 0.02, a constant transaction cost coefficientC = 0.01estimated from (14), and we assumed that the underlying asset pays no dividends, i.e.q = 0.In Fig. 3 we present results of calibration of implied couple(σRAPM , R). Interestinglyenough, two call options with higher strike pricesE = 25, 30 had almost constant impliedrisk premiumR. On the other the risk premium of an option with lowestE = 23 wasfluctuating and it had highest average ofR.

Finally, in Fig. 4 we present one week behavior of implied volatilities and risk premiumcoefficients for the Microsoft call option onE = 25 expiring atT = April 19, 2003. Inthe beginning of the investigated period the risk premium coefficientR was rather high and


Figure 4: One week behavior of Microsoft stocks (March 20 - 27, 2003) and call options withexpiration date April 19, 2003. Computed implied volatilitiesσRAPM and risk premiumsR.

fluctuating. On the other hand, it tends to a flat value ofR ≈ 5 at the end of the week.Interesting feature can be observed at the end of the second day when both stock and optionprices went suddenly down. The time series analysis of the implied volatility σRAPM fromfirst two days was unable to predict such a behavior. On the other, high fluctuation in theimplied risk premiumR during first two days can send a signal to an investor that suddenchanges can be expected in the near future.

3 Transformation method for a linear Black–Scholes equation

One of the interesting problems in this field is the analysis of the early exercise boundaryand the optimal stopping time for American options on stockspaying a continuous dividend.It can be easily reduced to a problem of solving a certain freeboundary problem for theBlack–Scholes equation (cf. Black & Scholes [7]). However,the exact analytical expressionfor the free boundary profile is not known yet. Many authors have investigated variousapproximate models leading to approximate expressions forvaluing American call and putoptions (see e.g. [24, 25, 33, 35, 36, 44, 45] and recent papers by Zhu [55, 56], Alobaidietal. [1,41,42], Stamicaret al.[50] and the survey paper by Chadam [12] and other referencestherein). For the purpose of studying the free boundary profile near expiry, many differentintegral equations have been derived [6,11,30,40].

Let us recall that the equation governing the time evolutionof the priceV (S, t) of the

14 DanielSevcovic

American call option is the following parabolic PDE:

∂V

∂t+ (r − q)S

∂V

∂S+

σ2

2S2 ∂2V

∂S2− rV = 0 , 0 < S < Sf (t), 0 < t < T ,

V (0, t) = 0, V (Sf (t), t) = Sf (t) − E ,∂V

∂S(Sf (t), t) = 1 , (23)

V (S, T ) = max(S − E, 0) ,

defined on a time-dependent domainS ∈ (0, Sf (t)), wheret ∈ (0, T ). As usualS > 0is the stock price,E > 0 is the exercise price,r > 0 is the risk-free rate,q > 0 is thecontinuous stock dividend rate and

σ ≡ const > 0

is a constant volatility of the underlying stock process.The main purpose of this section is to present an alternativeintegral equation which

will provide an accurate numerical method for calculating the early exercise boundary nearexpiry. The derivation of the nonlinear integral equation is based on the Fourier transform.A solution to this integral equation is the free boundary profile. The novelty of this approachconsists in three steps:

1. The fixed domain transformation.

2. Derivation of a parabolic PDE for the so-called syntheticportfolio with a nonlin-ear nonlocal constraint between the free boundary positionand the solution of thisparabolic equation itself.

3. Construction of a solution by means of Fourier sine and cosine integral transforms.

Throughout this section we restrict our attention to the case whenr > q > 0. It is wellknown that, forr > q > 0, the free boundary(τ) = Sf (T − τ) starts at (0) = rE/q,whereas (0) = E for the caser ≤ q (cf. Dewynneet al. [16], Kwok [38]). Thus, thefree boundary profile develops an initial jump in the caser > q > 0. Notice that the case0 < r ≤ q can be also treated by other methods based on integral equations. Kwok [38]derived another integral equation which covers both cases0 < r ≤ q, as well asr > q > 0(see equation (45) and Remark 3.2). However, in the latter case equation (45) becomessingular ast → T−, leading to numerical instabilities near expiry.

In the rest of this section to investigate the behavior of thefree boundarySf (t). Wepresent a method developed bySevcovic in [46] of reducing the free boundary problemfor (23) to a nonlinear integral equation with a singular kernel. Notice that our method ofreducing the free boundary problem to a nonlinear integral equation can be also successfullyadopted for valuing the American put option paying no dividends (cf. Stamicaret al. [50]).


3.1 Fixed domain transformation of the free boundary problem

In this section we will perform a fixed domain transformationof the free boundary problem(23) into a parabolic equation defined on a fixed spatial domain. As it will be shown below,imposing of the free boundary condition will result in a nonlinear time-dependent terminvolved in the resulting equation. To transform equation (23) defined on a time dependentspatial domain(0, Sf (t)), we introduce the following change of variables:

τ = T − t, x = ln

(

(τ)

S

)

where (τ) = Sf (T − τ). (24)

Clearly,τ ∈ (0, T ) andx ∈ (0,∞) wheneverS ∈ (0, Sf (t)). Let us furthermore define theauxiliary functionΠ = Π(x, τ) as follows:

Π(x, τ) = V (S, t) − S∂V

∂S(S, t). (25)

Notice that the quantityΠ has an important financial meaning as it is a synthetic portfolioconsisting of one long option and∆ = ∂V

∂S underlying short positioned stocks. It followsfrom (24) that

∂Π

∂x= S2 ∂2V

∂S2,

∂2Π

∂x2+ 2

∂Π

∂x= −S3 ∂3V

∂S3,

∂Π

∂τ+

˙

∂Π

∂x= S

∂2V

∂S∂t− ∂V

∂t, (26)

where ˙ = d/dτ . Now assuming thatV = V (S, t) is a sufficiently smooth solution of(23), we may differentiate (23) with respect toS. Plugging expressions (26) into (23), wefinally obtain that the functionΠ = Π(x, τ) is a solution of the parabolic equation

∂Π

∂τ+ a(τ)

∂Π

∂x− σ2

2

∂2Π

∂x2+ rΠ = 0, (27)

x ∈ (0,∞), τ ∈ (0, T ), with a time-dependent coefficient

a(τ) =˙(τ)

(τ)+ r − q − σ2

2.

It follows from the boundary conditionV (Sf (t), t) = Sf (t)−E andVS(Sf (t), t) = 1 that

Π(0, τ) = −E, Π(+∞, τ) = 0 . (28)

The initial conditionΠ(x, 0) can be deduced from the pay-off diagram forV (S, T ). Weobtain

Π(x, 0) =

{

−E for x < ln(

(0)E

)

0 otherwise.(29)

16 DanielSevcovic

Notice that equation (27) is a parabolic PDE with a time-dependent coefficienta(τ). Inwhat follows, we will show the functiona(τ) depends upon a solutionΠ itself. This depen-dence is non-local in the spatial variablex. Moreover, the initial position of the interface(0) enters the initial conditionΠ(x, 0). Therefore we have to determine the relationshipbetween the solutionΠ(x, τ) and the free boundary function(τ) first. To this end, wemake use of the boundary condition imposed onV at the interfaceS = Sf (t). SinceSf (t) − E = V (Sf (t), t) we have

d

dtSf (t) =

∂V

∂S(Sf (t), t)

d

dtSf (t) +

∂V

∂t(Sf (t), t) .

As ∂V∂S (Sf (t), t) = 1 we obtain ∂V

∂t (S, t) = 0 at S = Sf (t). Assuming the functionΠx has a trace atx = 0, and taking into account (26), we may conclude that, for anyt = T − τ ∈ [0, T ),

S2 ∂2V

∂S2(S, t) → ∂Π

∂x(0, τ), S

∂V

∂S(S, t) → (τ) asS → Sf (t)− .

If ∂V∂t (S, t) → ∂V

∂t (Sf (t), t) = 0 asS → Sf (t)−, then it follows from the Black–Scholesequation (23) that

(r − q)(τ) +σ2

2

∂Π

∂x(0, τ) − r((τ) − E)

= limS→Sf(t)−

(

∂V

∂t(S, t) + (r − q)S

∂V

∂S(S, t) +

σ2

2S2 ∂2V

∂S2(S, t) − rV (S, t)

)

= 0.

As a consequence we obtain a nonlocal algebraic constraint between the free boundaryfunction(τ) and the boundary trace∂xΠ(0, τ) of the derivative of the solutionΠ itself:

(τ) =rE

q+

σ2

2q

∂Π

∂x(0, τ) for 0 < τ ≤ T. (30)

It remains to determine the initial position of the interface (0). According to Dewynneetal. [16] (see also Kwok [38]), the initial position(0) of the free boundary isrE/q for thecase0 < q < r. Alternatively, we can derive this condition from (27)–(29) by assuming thesmoothness ofΠ in thex variable up to the boundaryx = 0 uniformly for τ → 0+. Moreprecisely, we assume that

limτ→0+

∂Π

∂x(0, τ) = lim

τ→0+,x→0+

∂Π

∂x(x, τ) = lim

x→0+

∂Π

∂x(x, 0) = 0

becauseΠ(x, 0) = −E for x close to0+. By (30) we obtain

(0) =rE

q. (31)


In summary, we have shown that, under suitable regularity assumptions imposed on a solu-tion Π to (27), (28), (29), the free boundary problem (23) can be transformed into the initialboundary value problem for parabolic PDE

∂Π

∂τ+ a(τ)

∂Π

∂x− σ2

2

∂2Π

∂x2+ rΠ = 0,

Π(0, τ) = −E, Π(+∞, τ) = 0, x > 0, τ ∈ (0, T ), (32)

Π(x, 0) =

{

−E for x < ln (r/q) ,0 otherwise,

wherea(τ) = ˙(τ)(τ) + r − q − σ2

2 and

(τ) =rE

q+

σ2

2q

∂Π

∂x(0, τ), (0) =

rE

q. (33)

We emphasize that the problem (32) constitutes a nonlinear parabolic equation with a non-local constraint given by (33).

Remark 3.1 In our derivation of the free boundary function(τ) and its initial condition(0) we did not assume that the solutionV (S, t) is C2 smooth up to the free boundaryS =Sf (t). Such an assumption would lead to an obvious contradictionΓ := ∂2V/∂S2 = 0 atS = Sf (t). On the other hand, the jump inΓ at the free boundary is the only driving forcefor the evolution of the free boundary function (see (30)). Construction of a PDE for thesynthetic portfolio functionΠ is a crucial step in our approach because the derivative∂xΠadmits a trace at the boundaryx = 0 and the unknown free boundary function can bedetermined via (33)

3.2 Reduction to a nonlinear integral equation

The main purpose of this section is to show how the fully nonlinear nonlocal problem (32)–(33) can reduced to a single nonlinear integral equation for(τ) giving the explicit formulafor the solutionΠ(x, τ) to (32). The idea is to apply the Fourier sine and cosine integraltransforms (cf. Stein & Weiss [51]). Let us recall that for any Lebesgue integrable functionf ∈ L1(R+) the sine and cosine transformations are defined as follows:FS(f)(ω) =∫∞0 f(x) sin ωxdx, FC(f)(ω) =

∫∞0 f(x) cos ωxdx. Their inverse transforms are given

by F−1S (g)(x) = 2

π

∫∞0 g(ω) sin ωxdω, F−1

C (g)(x) = 2π

∫∞0 g(ω) cos ωxdω. Now we

suppose that the function(τ) and subsequentlya(τ) are already know. LetΠ = Π(x, τ)be a solution of (32) corresponding to a given functiona(τ). Let us denote

p(ω, τ) = FS(Π(., τ))(ω), q(ω, τ) = FC(Π(., τ))(ω) (34)

whereω ∈ R+, τ ∈ (0, T ). By applying the sine and cosine integral transforms, taking

into account their basic properties and (33), we finally obtain a linear non-autonomousω-

18 DanielSevcovic

parameterized system of ODEs

d

dτp(ω, τ) − a(τ)ωq(ω, τ) + α(ω)p(ω, τ) = −Eω

σ2

2,

d

dτq(ω, τ) + a(τ)ωp(ω, τ) + α(ω)q(ω, τ) = −Ea(τ) − q(τ) + rE, (35)

for the sine and cosine transforms ofΠ (see (34)) where

α(ω) =1

2(σ2ω2 + 2r).

The system of equations (35) is subject to initial conditions at τ = 0, p(ω, 0) =FS(Π(., 0)(ω), q(ω, 0) = FC(Π(., 0)(ω). In the case of the call option, we deduce fromthe initial condition forΠ (see (32)) that

p(ω, 0) =E

ω

(

cos

(

ω lnr

q

)

− 1

)

, q(ω, 0) = −E

ωsin

(

ω lnr

q

)

. (36)

Taking into account (36) and by using the variation of constants formula for solving linearnon-autonomous ODEs, we obtain an explicit formula forp(ω, τ) = −Eω−1 + p(ω, τ)where

p(ω, τ) =E

ωe−α(ω)τ cos(ω(A(τ, 0) + ln(r/q)))

+

∫ τ

0e−α(ω)(τ−s)

[

rE

ωcos(ωA(τ, s)) + (rE − q(s)) sin(ωA(τ, s))

]

ds. (37)

Here we have denoted byA the function defined as

A(τ, s) =

∫ τ

sa(ξ) dξ = ln

(τ)

(s)+

(

r − q − σ2

2

)

(τ − s) . (38)

As F−1S (ω−1) = 1 we haveΠ(x, τ) = F−1

S (p(ω, τ)) = −E + 2π

∫∞0 p(ω, τ) sin(ωx) dω.

From (33) we conclude that the free boundary function satisfies the following equation:

(τ) =rE

q+

σ2

qπ

∫ ∞

0ωp(ω, τ) dω. (39)

Taking into account (37) and (39) we end up with the followingnonlinear singular integralequation for the free boundary function(τ):

(τ) =rE

q

(

1 +σ

r√

2πτexp

(

−rτ − (A(τ, 0) + ln(r/q))2

2σ2τ

)

(40)

+1√2π

∫ τ

0

[

σ +1

σ

(

1 − q(s)

rE

)

A(τ, s)

τ − s

]exp(

−r(τ − s) − A(τ,s)2

2σ2(τ−s)

)

√τ − s

ds

)

,


where the functionA depends upon via equation (38). To simplify this integral equation,we introduce a new auxiliary functionH : [0,

√T ] → R as follows:

(τ) =rE

q

(

1 + σ√

2H(√

τ))

. (41)

Using the change of variabless = ξ2 cos2 θ, one can rewrite the integral equation (39) interms of the functionH as follows:

H(ξ) = fH(ξ) +1√π

∫ π2

0[ξ cos θ − 2cotgθ H(ξ cos θ)gH(ξ, θ)] e−rξ2 sin2 θ−g2

H(ξ,θ) dθ,

(42)where

gH(ξ, θ) =1

σ√

2

1

ξ sin θln

(

1 + σ√

2H(ξ)

1 + σ√

2H(ξ cos θ)

)

+Λ√2ξ sin θ (43)

for ξ ∈ [0,√

T ], θ ∈ (0, π/2),

Λ =r − q

σ− σ

2

and

fH(ξ) =1

2r√

πξe−rξ2−

“

gH(ξ, π2)+ 1

ξ1

σ√

2ln(r/q)

”2

. (44)

Notice that equations (40) and (42) are integral equations with a singular kernel (cf. Gripen-berget al. [27]).

Remark 3.2 Kwok [38] derived another integral equation for the early exercise boundaryfor the American call option on a stock paying continuous dividend. According to Kwok [38,Section 4.2.3], (τ) satisfies the integral equation

(τ) = E + (τ)e−qτN(d) − Ee−rτN(d − σ√

τ)

+

∫ τ

0q(τ)e−qξN(dξ) − rEe−rξN(dξ − σ

√

ξ)dξ (45)

where

d =1

σ√

τln

(

(τ)

E

)

+ Λ√

2τ , dξ =1

σ√

ξln

(

(τ)

(τ − ξ)

)

+ Λ√

2ξ

and N(u) is the cumulative distribution function for the normal distribution. The aboveintegral equation covers both cases:r ≤ q as well asr > q. However, in the caser > qthis equation becomes singular asτ → 0+.

In the rest of this section we derive a formula for pricing American call options basedon the solution to the integral equation (42). With regard to (25), we have

∂

∂S

(

S−1V (S, t))

= −S−2Π(

ln(

S−1(T − t))

, T − t)

.

20 DanielSevcovic

Taking into account the boundary conditionV (Sf (t), t) = Sf (t) − E and integrating theabove equation fromS to Sf (t) = (T − t), we obtain

V (S, T − τ) =S

(τ)

(

(τ) − E +

∫ ln (τ)S

0exΠ(x, τ) dx

)

. (46)

It is straightforward to verify thatV given by (46 is indeed a solution to the free bound-ary problem (23). Inserting the expressions (37) and (39) (recall thatΠ(x, τ) = −E +2π

∫∞0 p(ω, τ) sin ωxdω) into (46), we end up with the formula for pricing the American

call option:

V (S, T − τ) = S − E +S

(τ)E I2(A(τ, 0) + ln(r/q), ln((τ)/S), τ)

+S

(τ)

∫ τ

0

[

rE I2(A(τ, s), ln((τ)/S), τ − s) (47)

+(rE − q(s)) I1(A(τ, s), ln((τ)/S), τ − s)

]

ds

for anyS ∈ (0, Sf (t)) andt ∈ [0, T ], whereA(τ, s) = ln (τ)(s) + (r− q− σ2

2 )(τ − s). Here

I1(A,L) =e−(r−σ2/2)τ

2

[

eAM

(−A − σ2τ

σ√

2τ,

L

σ√

2τ

)

− e−AM

(

A − σ2τ

σ√

2τ,

L

σ√

2τ

)]

,

I2(A,L) =e−rτeL

2M

(

A − L

σ√

2τ,

2L

σ√

2τ

)

(48)

−e−(r−σ2/2)τ

2

[

eAM

(−A − σ2τ

σ√

2τ,

L

σ√

2τ

)

+ e−AM

(

A − σ2τ

σ√

2τ,

L

σ√

2τ

)]

and

M(x, y) = erf(x + y) − erf(x) =2√π

∫ x+y

xe−ξ2

dξ.

We will refer to (47) as the semi-explicit formula for pricing the American call option.We use the term ‘semi-explicit’ because (47) contains the free boundary function(τ) =Sf (T − τ) which has to be determined first by solving the nonlinear integral equation (42).

3.3 Numerical experiments

In this section we focus on numerical experiments. We will compute the free boundaryprofile

Sf (t) = (T − t) =rE

q

(

1 + σ√

2H(√

T − t))

(49)

(see (36)) by solving the nonlinear integral equation (42).We also present a comparison ofthe results obtained by our methods to those obtained by other known methods for solvingthe American call option problem.


The computation of a solution of the nonlinear integral equation is based on an iterativemethod. We will construct a sequence of approximate solutions to (42). LetH0 be an initialapproximation of a solution to (42). If we supposeH0(ξ) = h1ξ, then, plugging this ansatzinto (42) yields the well-known first order approximation ofa solutionH(ξ) in the form

H0(ξ) = 0.451381 ξ

i.e. 0(τ) = rEq (1 + 0.638349σ

√τ) (cf. Sevcovic [46]). This asymptotics is in agreement

with that of Dewynneet al. [16]. Forn = 0, 1, ... we will define then + 1 approximationHn+1 as follows:

Hn+1(ξ) =

fHn(ξ) +1√π

∫ π2

0[ξ cos θ − 2cotgθ Hn(ξ cos θ)gHn(ξ, θ)] e−rξ2 sin2 θ−g2

Hn(ξ,θ) dθ (50)

for ξ ∈ [0,√

T ]. With regard to (43) and (44), we have

gHn(ξ, θ) =1

σ√

2

1

ξ sin θln

(

1 + σ√

2Hn(ξ)

1 + σ√

2Hn(ξ cos θ)

)

+Λ√2ξ sin θ

fHn(ξ) =1

2r√

πξe−rξ2−

“

gHn(ξ, π2)+ 1

ξ1

σ√

2ln

“

rq

””2

.

Notice that the functiongH is bounded provided thatH is nonnegative and Lipschitz con-tinuous on[0,

√T ]. Recall that we have assumedr > q > 0. Then the functionξ 7→ fH(ξ)

is bounded forξ ∈ [0,√

T ] and it vanishes atξ = 0. Moreover, ifH is smooth thenfH

is a flat function atξ = 0, i.e. fH(ξ) = o(ξn) as ξ → 0+ for all n ∈ N . From thenumerical point of view such a flat function can be omitted from computations. For smallvalues ofθ we approximate the function cotgθ gHn(ξ, θ) by its limit asθ → 0+. It yieldsthe approximation of the singular term in (50) in the form

cotgθgH(ξ, θ) ≈ 1

2

H ′(ξ)

1 + σ√

2H(ξ)+

Λ√2ξ for 0 < θ ≪ 1,

whereH = Hn. In the following numerical simulations we choseε ≈ 10−5.In what follows, we present several computational examples. In Fig. 5 we show five iter-

ates of the free boundary functionSf (t), where the auxiliary functionH(ξ) is constructedby means of successive iterations of the nonlinear integraloperator defined by the right-hand side of (42). This sequence converges to a fixed point of such a map, i.e. to a solutionof (42). Parameter values were chosen asE = 10, r = 0.1, σ = 0.2, q = 0.05, T = 1.Fig. 5 depicts the final tenth iteration of approximation of the functionSf (t). The meshcontained 100 grid points. One can observe very rapid convergence of iterates to a fixedpoint. In practice, no more than six iterates are sufficient to obtain the fixed point of (42).It is worth noting that in all our numerical simulations the convergence was monotone, i.e.the curve moves only up in the iteration process.

22 DanielSevcovic

Figure 5:Five successive approximations of the free boundarySf (t) obtained from equation (42).The profile of the solutionSf (t).

Figure 6:Long-time behavior ofSf (t).

In Fig. 6 we show the long time behavior of the free boundarySf (t), t ∈ [0, T ], forlarge values of the expiration timeT . For the parameter valuesT = 50, r = 0.1, q =0.05, σ = 0.35 andE = 10 the theoretical value ofSf (+∞) is 36.8179 (see Dewynneetal. [16]).

In Table 1 we show a comparison of results obtained by our method based on the semi-explicit formula (47) and those obtained by known methods based on trinomial trees (bothwith the depth of the tree equal to 100), finite difference approximation (with 200 spatialand time grids) and analytic approximation of Barone-Adesi& Whaley (cf. [6], [30, Ch.15, p. 384]), resp. It also turned out that the method based onsolving the integral equation(42) is 5-10 times faster than other methods based on trees orfinite differences. The reasonis that the computation ofV (S, t) for a wide range of values ofS based on the semi-explicitpricing formula (47) is very fast provided that the free boundary function has already beencomputed.

In Fig. 7 the early exercise boundarySf (t) is computed for various values of the pa-rameterq. In these computations we choseE = 10, T = 0.01, σ = 0.45. Of interest is thecase whereq is close tor (q = 0.09 andr = 0.1).


Table 1: Comparison of the method based on formula (48) with other methods for the parametervaluesE = 10, T = 1, σ = 0.2, r = 0.1, q = 0.05. The positionSf (0) = (T ) of the freeboundary att = 0 (i.e. atτ = T ) was computed asSf (0) = (T ) = 22.3754

Method\ The asset valueS 15 18 20 21 22.3754Our method 5.15 8.09 10.03 11.01 12.37Trinomial tree 5.15 8.09 10.03 11.01 12.37Finite differences 5.49 8.48 10.48 11.48 12.48Analytic approximation 5.23 8.10 10.04 11.02 12.38

Figure 7:The early exercise boundarySf (t) for r = 0.1, q = 0.05. The early exercise boundarySf (t) for r = 0.1, q = 0.09.

24 DanielSevcovic

3.4 Early exercise boundary for an American put option

In this section we present results of transformation methodapplied to valuation of the earlyexercise boundary for American style of a put option. Recallthat the early exercise bound-ary problem for American put option can be formulated as follows:

∂V

∂t+ rS

∂V

∂S+

σ2

2S2 ∂2V

∂S2− rV = 0 , 0 < t < T, Sf (t) < S < ∞ ,

V (+∞, t) = 0, V (Sf (t), t) = E − Sf (t) ,∂V

∂S(Sf (t), t) = −1 , (51)

V (S, T ) = max(E − S, 0) ,

defined on a time-dependent domainS ∈ (Sf (t),∞), wheret ∈ (0, T ) (cf. Kwok [38]).Again S > 0 stands for the stock price,E > 0 is the exercise price,r > 0 is the risk-freerate andσ > 0 is the volatility of the underlying stock process. We shall assume that theasset pay no dividends, i.e.q = 0. In order to perform a fixed domain transformation of thefree boundary problem (52) we introduce the following change of variables

x = ln

(

S

(τ)

)

, where τ = T − t, (τ) = Sf (T − τ).

Similarly as in the case of a call option we define a synthetised portfolio Π for the putoptionΠ(x, τ) = V − S ∂V

∂S . Then it is easy to verify thatΠ is a solution to the followingthe parabolic equation

∂Π

∂τ− a(τ)

∂Π

∂x− σ2

2

∂2Π

∂x2+ rΠ = 0, x > 0, τ ∈ (0, T ),

Π(0, τ) = E, Π(∞, τ) = 0, (52)

σ2

2

∂Π

∂x(0, τ) = −rE, for τ ∈ (0, T ), (53)

Π(x, 0) = 0 for x > 0,

wherea(τ) = ˙(τ)(τ)+r−σ2

2 (see Stamicaret al.[50]). Now, following the same methodologyof applying the Fourier transform we are able to derive an integral equation for the freeboundary position. It is just the equationσ

2

2∂Π∂x (0, τ) = −rE in which the left hand side is

expressed by the inverse Fourier transform of the solutionΠ as a weakly singular integraldepending on the free boundary position (see [50] for details). We omit the technicaldetails here. We just recall that the integral equation for the free boundary function(τ)yields the following expression for(τ)

(τ) = Ee−(r−σ2

2)τeσ

√2τη(τ)

in terms of a new auxiliary functionη(τ) (see Stamicaret al. [50] for details). Furtherasymptotic analysis of the integral equation enables us to derive an asymptotic formula for


0.0002 0.0004 0.0006 0.0008 0.001

9.825

9.85

9.875

9.9

9.925

9.95

9.975

0.00002 0.00004 0.00006 0.00008 0.0001

9.93

9.94

9.95

9.96

9.97

9.98

9.99

Figure 8: Left: asymptotic approximation vs. binomial method forσ = 0.25, r = 0.1, E = 10andT − t = 8.76 hrs. (0.001 of a year), MBW approximation vs. the asymptoticsolution (54) forT − t = 0.876 hrs (right).

η(τ) asτ → 0. Namely,

η(τ) ∼ −√

− ln

[

2r

σ

√2πτerτ

]

asτ → 0+. (54)

Next we examine how accurately our asymptotic approximation matches the data fromthe binomial method (cf. Kwok [38]). Near expiry at about onehour, the asymptotic ap-proximation matches the data from the binomial method (see Fig. 8). With σ = 0.25, r =0.1, E = 10 at 8.76 hours the approximation gives an overestimate but ofonly 0.4 cents(see [50]). We also compared our asymptotic solution with MacMillan, Barone-Adesi, andWhaley’s [6], [4, 384–386], [40] numerical approximation of the American put free bound-ary (see Fig. 8). They apply a transformation that results ina Cauchy-Euler equation thatcan be solved analytically. For times very close to expiry, one can see that our approxima-tion of the free boundary matches the data from the binomial and trinomial methods moreaccurately.

The numerical and analytical results obtained by the transformation method for solv-ing free boundary problem for American put options are in agreement with those obtainedrecently by Zhu [55,56], Mallieret al. [41,42], Kuske and Keller [36], Knessl [34].

4 Transformation method for a nonlinear Black–Scholes equa-tion

The main goal of this section is to perform a fixed domain transformation of the free bound-ary problem for the nonlinear Black–Scholes equation (2) into a parabolic equation definedon a fixed spatial domain. For the sake of simplicity we will present a detailed derivation ofan equation only for the case of an American call option. Derivation of the correspondingequation for the American put option is similar. Throughoutthis section we shall assumethe volatility σ2 appearing in the Black–Scholes equation to be a function of the option

26 DanielSevcovic

priceS, time to expiryT − t andS2∂2SV , i.e.

σ = σ(S2∂2SV, S, T − t) .

Following (24) we shall consider the following change of variables:

τ = T − t, x = ln ((τ)/S) where (τ) = Sf (T − τ).

Thenτ ∈ (0, T ) andx ∈ (0,∞) iff S ∈ (0, Sf (t)). The boundary valuex = 0 correspondsto the free boundary positionS = Sf (t) whereasx ≈ +∞ corresponds to the default valueS = 0 of the underlying asset. Following Stamicaret al. [50] andSevcovic [46, 47] weconstruct the so-called synthetic portfolio functionΠ = Π(x, τ) defined as follows:

Π(x, τ) = V (S, t) − S∂V

∂S(S, t) . (55)

Again, it represents a synthetic portfolio consisting of one long positioned option and∆ =∂SV underlying short stocks. Similarly as in Section 3 we have

∂Π

∂x= S2 ∂2V

∂S2,

∂Π

∂τ+

˙

∂Π

∂x= − ∂

∂t

(

V − S∂V

∂S

)

where we have denoted = d/dτ . Assuming sufficient smoothness of a solutionV = V (S, t) to (2) we can deduce from (2) a parabolic equation for the synthetic port-folio function Π = Π(x, τ)

∂Π

∂τ+ (b(τ) − 1

2σ2)

∂Π

∂x− 1

2

∂

∂x

(

σ2 ∂Π

∂x

)

+ rΠ = 0

defined on a fixed domainx ∈ R, t ∈ (0, T ), with a time-dependent coefficient

b(τ) =˙(τ)

(τ)+ r − q (56)

and a diffusion coefficient given by:σ2 = σ2(∂xΠ(x, τ), (τ)e−x, τ) depending onτ, x and the gradient∂xΠ of a solutionΠ. Now the boundary conditionsV (0, t) =0, V (Sf (t), t) = Sf (t) − E and∂SV (Sf (t), t) = 1 imply

Π(0, τ) = −E, Π(+∞, τ) = 0 , 0 < τ < T , (57)

and, from the terminal pay-off diagram forV (S, T ), we deduce

Π(x, 0) =

{

−E for x < ln(

(0)E

)

0 otherwise.(58)

In order to close up the system of equations that determines the value of a syntheticportfolio Π we have to construct an equation for the free boundary position (τ). In-deed, both the coefficientb as well as the initial conditionΠ(x, 0) depend on the func-tion (τ). Similarly as in the case of a constant volatilityσ (see [46, 50]) we pro-ceed as follows: sinceSf (t) − E = V (Sf (t), t) and ∂SV (Sf (t), t) = 1 we have


ddt

Sf (t) = ∂SV (Sf (t), t) ddt

Sf (t) + ∂tV (Sf (t), t) and so∂tV (S, t) = 0 along the freeboundaryS = Sf (t). Moreover, assuming∂xΠ is continuous up to the boundaryx = 0 we obtainS2∂2

SV (S, t) → ∂xΠ(0, τ) andS∂SV (S, t) → (τ) asS → Sf (t)−.Now, by taking the limitS → Sf (t)− in the Black–Scholes equation (2) we obtain(r − q)(τ) + 1

2σ2∂xΠ(0, τ) − r((τ) − E) = 0. Therefore

(τ) =rE

q+

1

2qσ2(∂xΠ(0, τ), (τ), τ)

∂Π

∂x(0, τ)

for 0 < τ ≤ T . The value of (0) can be easily derived from the smoothness assumptionmade on∂xΠ at the originx = 0, τ = 0 under the structural assumption

0 < q < r (59)

made on the interest and dividend yield ratesr, q (cf. [46, 47]). Indeed, continuity of∂xΠat the origin(0, 0) implies limτ→0+ ∂xΠ(0, τ) = ∂xΠ(0, 0) = limx→0+ ∂xΠ(x, 0) = 0becauseΠ(x, 0) = −E for x close to0+ providedln(r/q) > 0. From the above equationfor (τ) we deduce (0) = rE

q by taking the limitτ → 0+. Putting all the above equationstogether we end up with a closed system of equations forΠ = Π(x, τ) and = (τ)

∂Π

∂τ+(

b(τ) − σ2

2

)∂Π

∂x− 1

2

∂

∂x

(

σ2 ∂Π

∂x

)

+ rΠ = 0 ,

Π(0, τ) = −E , Π(+∞, τ) = 0 , x > 0 , τ ∈ (0, T ) ,

Π(x, 0) =

{

−E for x < ln(r/q)0 otherwise ,

(60)

whereσ = σ(∂xΠ(x, τ), (τ)e−x, τ) , b(τ) = ˙(τ)(τ) + r − q and the free boundary position

(τ) = Sf (T − τ) satisfies an implicit algebraic equation

(τ) =rE

q+

σ2(∂xΠ(0, τ), (τ), τ)

2q

∂Π

∂x(0, τ) , with (0) =

rE

q, (61)

whereτ ∈ (0, T ). Notice that, in order to guarantee parabolicity of equation (60) we haveto assume that the functionp 7→ σ2(p, (τ)e−x, τ)p is strictly increasing. More precisely,we shall assume that there exists a positive constantγ > 0 such that

σ2(p, ξ, τ) + p∂pσ2(p, ξ, τ) ≥ γ > 0 (62)

for anyξ > 0, τ ∈ (0, T ) andp ∈ R. Notice that condition (62) is satisfied for the RAPM

model in whichσ2 = σ2(1 + µp13 ξ−

13 ) for anyµ ≥ 0 andp ≥ 0. Clearlyp = S2∂2

SV > 0for the case of plain vanilla call or put options. As far as theBarles and Soner model isconcerned, we haveσ2 = σ2(1 + Ψ(a2erτp)) and condition (62) is again satisfied becausethe functionΨ is a positive and increasing function in the Barles and Sonermodel.

28 DanielSevcovic

Remark 4.1 Following exactly the same argument as in (46) one can derivean explicitexpression for the option priceV (S, t):

V (S, T − τ) =S

(τ)

(

(τ) − E +

∫ ln(τ)

S

0exΠ(x, τ) dx

)

. (63)

4.1 An iterative algorithm for approximation of the early exercise boundary

The idea of the iterative numerical algorithm for solving the problem (60), (61) is rathersimple: we use the backward Euler method of finite differences in order to discretize theparabolic equation (60) in time. In each time level we find a new approximation of a solu-tion pair (Π, ). First we determine a new position of from the algebraic equation (61).We remind ourselves that (even in the caseσ is constant) the free boundary function(τ)behaves likerE/q + O(τ1/2) for τ → 0+ (see e.g. [16] or [46]) and sob(τ) = O(τ−1/2).Hence the convective termb(τ)∂xΠ becomes a dominant part of equation (60) for smallvalues ofτ . In order to overcome this difficulty we employ the operator splitting techniquefor successive solving of the convective and diffusion parts of equation (60). Since the dif-fusion coefficientσ2 depends on the derivative∂xΠ of a solutionΠ itself we make severalmicro-iterates to find a solution of a system of nonlinear algebraic equations.

Now we present our algorithm in more details. We restrict thespatial domainx ∈(0,∞) to a finite interval of valuesx ∈ (0, L) whereL > 0 is sufficiently large. For prac-tical purposes one can takeL ≈ 3 as it corresponds to the intervalS ∈ (Sf (t)e−L, Sf (t))in the original asset price variableS. The valueSf (t)e−L is then could be a good approx-imation for the default valueS = 0 if L ≈ 3. Let us denote byk > 0 the time step,k = T/m, and, byh > 0 the spatial step,h = L/n wherem,n ∈ N stand for thenumber of time and space discretization steps, resp. We denote byΠj

i an approximation ofΠ(xi, τj), j ≈ (τj), bj ≈ b(τj) wherexi = ih, τj = jk. We approximate the value ofthe volatility σ at the node(xi, τj) by finite difference as follows:

σji = σj

i (j ,Πj) = σ((Πj

i+1 − Πji )/h, je−xi , τj) .

Then for the Euler backward in time finite difference approximation of equation (60) wehave

Πj − Πj−1

k+

(

bj − 1

2(σj)2

)

∂xΠj − 1

2∂x

(

(σj)2∂xΠj)

+ rΠj = 0 (64)

and the solutionΠj = Πj(x) is subject to Dirichlet boundary conditions atx = 0 andx = L. We setΠ0(x) = Π(x, 0). Now we decompose the above problem into two parts - aconvection part and a diffusive part by introducing an auxiliary intermediate stepΠj− 1

2 :

(Convective part)

Πj− 12 − Πj−1

k+ bj∂xΠj− 1

2 = 0 , (65)


(Diffusive part)

Πj − Πj− 12

k− (σj)2

2∂xΠj − 1

2∂x

(

(σj)2∂xΠj)

+ rΠj = 0 . (66)

The idea of the operator splitting technique consists in comparison the sum of solutions toconvective and diffusion part to a solution of (64). Indeed,if ∂xΠj ≈ ∂xΠj− 1

2 then it isreasonable to assume thatΠj computed from the system (65)–(66) is a good approximationof the system (64).

The convective part can be approximated by an explicit solution to the transport equa-tion:

∂τ Π + b(τ)∂xΠ = 0 for x > 0, τ ∈ (τj−1, τj ] (67)

subject to the boundary conditionΠ(0, τ) = −E and initial conditionΠ(x, τj−1) =Πj−1(x). For American style of call option the free boundary(τ) = Sf (T − τ) mustbe an increasing function inτ and we have assumed0 < q < r we haveb(τ) =˙(τ)/(τ)+r−q > 0 and so prescribing the in-flowing boundary conditionΠ(0, τ) = −Eis consistent with the transport equation. Let us denote byB(τ) the primitive function tob(τ), i.e. B(τ) = ln (τ) + (r − q)τ . Equation (67) can be integrated to obtain its explicitsolution:

Π(x, τ) =

{

Πj−1(x − B(τ) + B(τj−1)) if x − B(τ) + B(τj−1) > 0 ,−E otherwise.

(68)

Thus the spatial approximationΠj− 1

2i can be constructed from the formula

Πj− 1

2i =

{

Πj−1(ξi) if ξi = xi − ln j + ln j−1 − (r − q)k > 0 ,−E otherwise,

(69)

where a linear approximation between discrete valuesΠj−1i , i = 0, 1, ..., n, is being used to

compute the valueΠj−1(xi − ln j + ln j−1 − (r − q)k).The diffusive part can be solved numerically by means of finite differences. Using

central finite difference for approximation of the derivative ∂xΠj we obtain

Πji − Π

j− 12

i

k+ rΠj

i − (σji )

2

2

Πji+1 − Πj

i−1

2h

− 1

2h

(

(σji )

2 Πji+1 − Πj

i

h− (σj

i−1)2 Πj

i − Πji−1

h

)

= 0 .

Hence, the vector of discrete valuesΠj = {Πji , i = 1, 2, ..., n} at the time levelj ∈

{1, 2, ...,m} satisfies the tridiagonal system of equations

αjiΠ

ji−1 + βj

i Πji + γj

i Πji+1 = Π

j− 12

i , (70)

30 DanielSevcovic

for i = 1, 2, ..., n, where

αji ≡ αj

i (j ,Πj) = − k

2h2(σj

i−1)2 +

k

2h

(σji )

2

2,

γji ≡ γj

i (j ,Πj) = − k

2h2(σj

i )2 − k

2h

(σji )

2

2, (71)

βji ≡ βj

i (j ,Πj) = 1 + rk − (αj

i + γji ) .

The initial and boundary conditions atτ = 0 andx = 0, L, resp., can be approximated asfollows:

Π0i =

{

−E for xi < ln (r/q) ,0 for xi ≥ ln (r/q) ,

for i = 0, 1, ..., n, andΠj0 = −E, Πj

n = 0.Next we proceed by approximation of equation (61) which introduces a nonlinear con-

straint condition between the early exercise boundary function (τ) and the trace of thesolutionΠ at the boundaryx = 0 (S = Sf (t) in the original variable). Taking a finitedifference approximation of∂xΠ at the originx = 0 we obtain

j =rE

q+

1

2qσ2(

(Πj1 − Πj

0)/h, j , τj

) Πj1 − Πj

0

h. (72)

Now, equations (69), (70) and (72) can be written in an abstract form as a system ofnonlinear equations:

j = F(Πj , j),

Πj− 12 = T (Πj , j), (73)

A(Πj, j)Πj = Πj− 12 ,

whereF(Πj , j) is the right-hand side of the algebraic equation (72),T (Πj , j) is thetransport equation solver given by the right-hand side of (69) andA = A(Πj, j) is a tridi-agonal matrix with coefficients given by (71). The system (73) can be approximately solvedby means of successive iterates procedure. We define, forj ≥ 1, Πj,0 = Πj−1, j,0 = j−1.Then the(p + 1)-th approximation ofΠj andj is obtained as a solution to the system:

j,p+1 = F(Πj,p, j,p),

Πj− 12,p+1 = T (Πj,p, j,p+1), (74)

A(Πj,p, j,p+1)Πj,p+1 = Πj− 12,p+1 .

Notice that the last equation is a linear tridiagonal equation for the vectorΠj,p+1 whereasj,p+1 andΠj− 1

2,p+1 can be directly computed from (72) and (69), resp. If the sequence of

approximate solutions{(Πj,p, j,p)}∞p=1 converges to some limiting value(Πj,∞, j,∞) asp → ∞ then this limit is a solution to a nonlinear system of equations (73) at the time levelj and we can proceed by computing the approximate solution thenext time levelj + 1.


4.2 Numerical approximations of the early exercise boundary

In this section we focus on numerical experiments based on the iterative scheme describedin the previous section. The main purpose is to compute the free boundary profileSf (t) =(T − t) for different (non)linear Black–Scholes models and for various model parameters.A solution (Π, ) has been computed by our iterative algorithm for the following basicmodel parameters:E = 10, T = 1 (one year),r = 0.1 (10% p.a) ,q = 0.05 (5% p.a.) andσ = 0.2. We usedn = 750 spatial points andm = 225000 time discretization steps. Sucha time stepk = T/m corresponds to 140 seconds between consecutive time levelswhenexpressed in real time scale. In average we neededp ≤ 6 micro-iterates (74) in order tosolve the nonlinear system (73) with the precision10−7.

4.2.1 Case of a constant volatility – comparison study

In our first numerical experiment we make attempt to compare our iterative approxima-tion scheme for solving the free boundary problem for an American call option to knownschemes in the case when the volatilityσ > 0 is constant. We compare our solutionto the one computed by means of a solution to a nonlinear integral equation for(τ)(see also [46, 50]). This comparison can be also considered as a benchmark or test ex-ample for which we know a solution that can be computed by a another justified algo-rithm. In Fig. 9, part a), we show the functioncomputed by our iterative algorithm forE = 10, T = 1, r = 0.1, q = 0.05, σ = 0.2. At the expiryT = 1 the value of (T ) wascomputed as: (T ) = 22.321. The corresponding value(T ) computed from the integralequation (40) (cf. [46]) was(T ) = 22.375. The relative error is less than 0.25%. In thepart b) we present 7 approximations of the free boundary function (τ) computed for dif-ferent mesh sizesh (see Tab. 2 for details). The sequence of approximate free boundariesh, h = h1, h2, ..., converges monotonically from below to the free boundary function ash ↓ 0. The next part c) of Fig. 9 depicts various solution profiles of a functionΠ(x, τ).In order to achieve a reasonable approximation to equation (72) we need very accurate ap-proximation ofΠ(x, τ) for x close to the origin0. The parts d) and e) of Fig. 9 depict thecontour and 3D plots of the functionΠ(x, τ).

In Tab. 2 we present the numerical error analysis for the distance‖h−‖p measured intwo different norms (L∞ andL2) of a computed free boundary positionh correspondingto the mesh sizeh and the solution computed from the integral equation described in (40)(cf. [46]). The time stepk has been adjusted to the spatial mesh sizeh in order to satisfyCFL conditionσ2k/h2 ≈ 1/2. We also computed the experimental order of convergenceeoc(Lp) for p = 2,∞. Recall that the experimental order of convergence can be defined asthe ratio:

eoc(Lp) =ln(‖hi

− ‖p) − ln(‖hi−1− ‖p)

lnhi − ln hi−1.

It can be interpreted as an exponentα = eoc(Lp) for which we have‖h − ‖p = O(hα).It turns out from Tab. 2 that the conjecture on the order of convergence‖h − ‖∞ = O(h)whereas‖h − ‖2 = O(h3/2) ash → 0+ could be reasonable.

32 DanielSevcovic

a) b)

c)

d) e)

Figure 9:a) A comparison of the free boundary function(τ) computed by the iterative algorithm(green solid curve) to the integral equation based approximation (dashed red curve); b) free boundarypositions computed for various mesh sizes; c) a solution profile Π(x, τ) for τ = 0 (blue line),τ = T/2 (red curve),τ = T (green curve); d) 3D plot and e) contour plot of the functionΠ(x, τ).


Table 2: Experimental order of convergence of the iterative algorithm for approximating the freeboundary position.

h err(L∞) eoc(L∞) err(L2) eoc(L2)

0.03 0.5 - 0.808 -0.012 0.215 0.92 0.227 1.390.006 0.111 0.96 0.0836 1.440.004 0.0747 0.97 0.0462 1.460.003 0.0563 0.98 0.0303 1.470.0024 0.0452 0.98 0.0218 1.480.002 0.0378 0.98 0.0166 1.48

Figure 10:A comparison of the free boundary functionR(τ) computed for the Risk Adjusted Pric-ing Methodology model. Dashed red curve represents a solution corresponding toR = 0, whereasthe green curves represent a solutionR(τ) for different values of the risk premium coefficientsR = 5, 15, 40, 70, 100.

4.2.2 Risk Adjusted Pricing Methodology model

In the next example we computed the position of the free boundary (τ) in the case of theRisk Adjusted Pricing Methodology model - a nonlinear Black–Scholes type model derivedby Jandacka andSevcovic in [32]. In this model the volatilityσ is a nonlinear function ofthe asset priceS and the second derivative∂2

SV of the option price, and it is given byformula (8). In Fig. 10 we present results of numerical approximation of the free boundarypositionR(τ) = SR

f (T −τ) in the case when the coefficient of transaction costsC = 0.01is fixed and the risk premium measureR varies fromR = 5, 15, 40, 70, up toR = 100. Wecompare the position of the free boundaryR(τ) to the case when there are no transactioncosts and no risk from volatile portfolio, i.e. we compare itwith the free boundary position0(τ) for the linear Black–Scholes equation (see Fig. 10). An increase in the risk premiumcoefficientR resulted in an increase of the free boundary position as it can be expected.

In Tab. 3 and Fig. 10 we summarize results of comparison of thefree boundary positionR for various values of the risk premium coefficient to the reference position = 0

34 DanielSevcovic

Table 3:Distance‖R − 0‖p (p = 2,∞) of the free boundary positionR from the reference freeboundary position 0 and experimental ordersα∞ andα2 of convergence.

R ‖R − 0‖∞ α∞ ‖R − 0‖2 α2

1 0.0601 - 0.0241 -2 0.0754 0.33 0.0303 0.3285 0.102 0.33 0.0408 0.326

10 0.128 0.33 0.0511 0.32415 0.145 0.32 0.0582 0.32320 0.16 0.32 0.0639 0.32230 0.182 0.32 0.0727 0.32140 0.2 0.32 0.0798 0.3250 0.214 0.32 0.0856 0.31960 0.227 0.32 0.0907 0.31870 0.239 0.32 0.0953 0.31780 0.249 0.32 0.0994 0.31790 0.259 0.32 0.103 0.316

100 0.268 0.32 0.107 0.316

Figure 11:Dependence of the norms‖R − 0‖p (p = ∞, 2) of the deviation of the free boundary = R(τ) for the RAPM model on the risk premium coefficientR.


Figure 12: A comparison of the free boundary function(τ) computed for the Barles andSoner model. Dashed red curve represents a solution corresponding to R = 0, whereas thegreen curves represents a solution(τ) for different values of the risk aversion coefficienta =0.01, 0.07, 0.13, 0.25, 0.35.

Figure 13:Dependence of the norms‖a − 0‖p (p = ∞, 2) of the deviation of the free boundary = a(τ) for the Barles-Soner model on the risk aversion parametera.

computed from the Black–Scholes model with a constant volatility σ = σ, i.e. R = 0. Theexperimental orderαp of the distance function‖R − 0‖p = O(Rαp) has been computedfor p = 2,∞, as follows:

αp =ln(‖Ri − 0‖p) − ln(‖Ri−1 − 0‖p)

lnRi − ln Ri−1.

According to the values presented in Tab. 3 it turns out that areasonable conjecture on theorder of convergence is that‖R−0‖p = O(R1/3) for both normsp = 2 andp = ∞. Sincethe transaction cost coefficientC and risk premium measureR enter the expression for theRAPM volatility (8) only in the productC2R we can conjecture that‖R,C − 0,0‖p =O(C2/3R1/3) as eitherC → 0+ or R → 0+.

36 DanielSevcovic

Table 4:Distance‖a − 0‖p (p = 2,∞) of the free boundary positiona from the reference freeboundary position 0 and experimental ordersα∞ andα2 of convergence.

a ‖a − 0‖∞ α∞ ‖a − 0‖2 α2

0.01 0.156 - 0.0615 -0.02 0.25 0.68 0.0985 0.680.05 0.472 0.69 0.184 0.6790.07 0.602 0.72 0.232 0.690.1 0.793 0.77 0.298 0.7120.11 0.857 0.82 0.32 0.740.13 0.99 0.86 0.364 0.7660.15 1.13 0.92 0.409 0.8070.2 1.52 1. 0.529 0.8970.25 1.97 1.2 0.669 1.050.3 2.49 1.3 0.833 1.210.35 3.07 1.4 1.03 1.35

4.2.3 Barles and Soner model

Our next example is devoted to the nonlinear Black–Scholes model due to Barles and Soner(see [8]). In this model the volatility is given by equation (6). Numerical results are depictedin Fig. 12. Choosing a larger value of the risk aversion coefficienta > 0 resulted in increaseof the free boundary positiona(τ). The position of the early exercise boundarya(τ) hasconsiderably increased in comparison to the linear Black–Scholes equation with constantvolatility σ = σ. In contrast to the case of constant volatility as well as theRAPM model,there is, at least a numerical evidence (see Fig.12 anda for the largest valuea = 0.35) thatthe free boundary profilea(τ) need not be necessarily convex. Recall that that convexityof the free boundary profile has been proved analytically by Ekstromet al. and Chenet al.in a recent papers [13, 19, 20] in the case of a American put option and constant volatilityσ = σ.

Similarly as in the previous model we also investigated the dependence of the freeboundary position = a(τ) on the risk aversion parametera > 0. In Tab. 4 and Fig. 13we present results of comparison of the free boundary position a for various values of therisk aversion coefficienta to the reference position = 0. Inspecting valuesαp of theorder of distance‖a − 0‖p it can be conjectured that‖a − 0‖p = O(a2/3) asa → 0 forboth normsp = 2 andp = ∞.

5 Transformation methods for Asian call options

Path dependent options are options whose pay-off diagram depends on the path historyof the underlying asset. Among path dependent options Asianoptions plays an importantrole as they are quite common in currency and commodity markets like e.g. oil industry


(cf. [15, 28]). Asian options may depend on the averaged pathhistory in several ways. Weshall restrict our attention to the so-called floating strike Asian call options. The floatingstrike price is assumed to be an arithmetic average of the underlying asset prices over theentire time interval[0, T ] whereT is the expiration time.

Let us define the arithmetic averageA = At of the underlying assetS = St by

At =1

t

∫ t

0Sτ dτ .

For the case of the so-called Asian floating strike call option the pay-off diagram at expiryTreads as follows:V (S,A, T ) = max(S −A, 0). It means the priceV of an option contractwill depend not only on the underlying asset priceS, time t, but also on the underlyingasset path averageA, i.e. V = V (S,A, t).

5.1 Governing equations for Asian options

As it is usual in the option pricing theory, we shall describethe asset price dynamics by ageometric Brownian with drift , dividend yieldq ≥ 0 and volatility σ, i.e. dS = ( −q)Sdt + σSdW whereW is the standard Wiener process. If we apply Ito’s formula tothefunctionV = V (S,A, t) we obtain

dV =

(

∂V

∂t+

σ2

2S2 ∂2V

∂S2

)

+∂V

∂SdS +

∂V

∂AdA .

In the case of arithmetic averaging we havedA = t−1(S −A)dt. Hence the differentialdAis of the order ofdt and this is why the above expression fordV indeed represents its lowestorder approximation when taking into account stochastic character of the dynamics of theasset priceS. Therefore, following standard arguments from the Black–Scholes theory onecan derive the governing equation for pricing Asian option with arithmetic averaging in theform:

∂V

∂t+

σ2

2S2 ∂2V

∂S2+ S(r − q)

∂V

∂S+

S − A

t

∂V

∂A− rV = 0, (75)

where0 < t < T, S,A > 0 (see e.g. [15]). For Asian call option the above equation issubject to the terminal pay-off condition

V (S,A, T ) = max(S − A, 0), S,A > 0 . (76)

It is well known (see e.g. [15,38]) that for Asian options with floating strike we can achievedimension reduction by introducing the following similarity variable:

x =S

A, W (x, τ) =

1

AV (S,A, t)

whereτ = T − t. It is straightforward to verify thatV (S,A, t) = W (S/A, T − t)A is asolution of (75) iffW = W (x, τ) is a solution to the following parabolic PDE:

∂W

∂τ− σ2

2x2 ∂2W

∂x2− (r − q)x

∂W

∂x− x − 1

T − τ

(

W − x∂W

∂x

)

+ rW = 0, (77)

38 DanielSevcovic

wherex > 0 and0 < τ < T . The initial condition forW immediately follows from theterminal pay-off diagram for the call option,

W (x, 0) = max(x − 1, 0). (78)

5.2 American style of Asian options

Following Dai and Kwok [15], American style of Asian optionsis characterized by theexercise region

E = {(S,A, t) ∈ [0,∞) × [0,∞) × [0, T ), V (S,A, t) = V (S,A, T )}.

In the case of a call option this region can be described by an early exercise boundaryfunctionSf = Sf (A, t) such that

E = {(S,A, t) ∈ [0,∞) × [0,∞) × [0, T ), S ≥ Sf (A, t)}.

For American style of an Asian call option we have to impose a homogeneous Dirichletboundary conditionV (0, A, t) = 0 atS = 0. According to [15] theC1 continuity conditionat the point(Sf (A, t), A, t) of a contact of a solutionV with its pay-off diagram impliesthe following boundary condition at the free boundary position Sf (A, t):

∂V

∂S(Sf (A, t), A, t) = 1, V (Sf (A, t), A, t) = Sf (A, t) − A (79)

for anyA > 0 and0 < t < T . It is important to emphasize that the free boundary functionSf can be also reduced to a function of one variable by introducing a new state functionxf (t) as follows:

Sf (A, t) = Axf (t).

The functionxf = xf (t) is a free boundary function for the transformed state variablex = S/A. For American style of Asian call options the spatial domainfor the reducedequation (77) is given by

0 < x < (τ), τ ∈ (0, T ), where(τ) = xf (T − τ) .

Taking into account boundary conditions (79) for the optionpriceV we end up with corre-sponding boundary conditions for the functionW :

W (0, τ) = 0, W (x, τ) = x − 1,∂W

∂x(x, τ) = 1 at x = (τ) (80)

for any0 < τ < T and the initial condition

W (x, 0) = max(x − 1, 0) (81)

for anyx > 0.


5.3 Fixed domain transformation for American style of Asiancall options

Similarly as in Section 3, in order to apply the fixed domain transformation for the freeboundary problem (77), (80), (81) we introduce a new variable ξ and an auxiliary functionΠ = Π(ξ, τ) (again representing a synthetic portfolio) defined as follows:

ξ = ln

(

(τ)

x

)

, Π(ξ, τ) = W (x, τ) − x∂W

∂x. (82)

Clearly,x ∈ (0, (τ)) iff ξ ∈ (0,∞) for τ ∈ (0, T ). The valueξ = ∞ of the transformedvariable corresponds to the valuex = 0 (S = 0) expressed in the original variable. Onthe other hand, the valueξ = 0 corresponds to the free boundary positionx = (τ) (S =ASf (A, t)).

A straightforward calculation similar to that of Section 4 enables us to to justify that thefunctionΠ = Π(ξ, τ) is a solution to the following parabolic PDE

∂Π

∂τ+ a(ξ, τ)

∂Π

∂ξ− σ2

2

∂2Π

∂ξ2+

(

r +1

T − τ

)

Π = 0, (83)

where the terma(ξ, τ) is given by

a(ξ, τ) =ρ(τ)

ρ(τ)+ r − q − σ2

2− ρe−ξ − 1

T − τ. (84)

Notice the spatial dependence of the coefficienta = a(ξ, τ) in comparison to the case oftransformed equations for a linear or nonlinear Black–Scholes equations (see (32), (56)).The initial condition for the solutionΠ follows from (81)

Π(ξ, 0) =

{

−1 ξ < ln ρ(0),0 ξ > ln ρ(0).

Since∂xW (x, τ) = 1 for x = (τ) andW (0, τ) = 0 we conclude the Dirichlet boundaryconditions for the functionΠ

Π(0, τ) = −1, Π(∞, τ) = 0.

It remains to determine an algebraic constraint between thefree boundary function (τ)and the solutionΠ. Similarly as in the case of a linear or nonlinear Black–Scholes equationwe obtain, by differentiation the conditionW (ρ(τ), τ) = ρ(τ) − 1 with respect toτ, thefollowing identity:

d

dτρ(τ) =

∂W

∂x(ρ(τ), τ)

d

dτρ(τ) +

∂W

∂τ(ρ(τ), τ).

Since ∂W∂x (ρ(τ), τ) = 1 we have∂W

∂τ (x, τ) = 0 at x = (τ). Assuming continuity of thefunctionΠ and its derivativeΠξ up to the boundaryξ = 0 we have

x2 ∂2W

∂x2(x, τ) → ∂Π

∂ξ(0, τ), x

∂W

∂x(x, τ) → ρ(τ)

40 DanielSevcovic

asx → ρ(τ). Passing to the limitx → ρ(τ) in equation (77) we end up with the equation

−(r − q)ρ(τ) − σ2

2

∂Π

∂ξ(0, τ) +

ρ(τ) − 1

T − τ+ r[ρ(τ) − 1] = 0.

It yields an algebraic nonlocal expression for the free boundary position (τ)

ρ(τ) =r + 1

T−τ + σ2

2∂Π∂ξ (0, τ)

q + 1T−τ

. (85)

Next we determine the starting point of the free boundary function (0). It means that wehave to find the terminal value of the original state variablexf (T ). We shall assume astructural assumption

r > q ≥ 0 (86)

on the interest and dividend ratesr, q. If (0) > 1 thenln (0) > 0 and this is why the initialfunctionΠ(ξ, 0) is equal to−1 in some right neighborhood ofξ = 0. Thus∂ξΠ(ξ, 0) = 0for 0 < ξ ≪ 1. Again, assuming continuity of∂ξΠ(ξ, τ) at (ξ, τ) = (0, 0) we obtain

limτ→0+

∂Π

∂ξ(0, τ) = lim

τ→0+,x→0+

∂Π

∂ξ(x, τ) = lim

x→0+

∂Π

∂ξ(x, 0) = 0.

As a consequence we can conclude

ρ(0) =r + 1

T

q + 1T

=1 + rT

1 + qT> 1.

This initial condition for(0) is exactly the same as the one derived recently by Dai andKwok in [15]. They proved, for a general choice ofr, q ≥ 0 that the initial condition for thefunction is given by

(0) = max

(

1 + rT

1 + qT, 1

)

.

In summary, we have transformed the free boundary problem for pricing American style ofAsian call option with floating strike price into the following nonlocal parabolic PDE with


an algebraic constraint

∂Π

∂τ+ a(ξ, τ)

∂Π

∂ξ− σ2

2

∂2Π

∂ξ2+

(

r +1

T − τ

)

Π = 0, 0 < τ < T, ξ > 0,

ρ(τ) =1 + r(T − τ) + σ2

2 (T − τ)∂Π∂ξ (0, τ)

1 + q(T − τ), 0 < τ < T,

subject to initial and boundary conditions

Π(0, τ) = −1, Π(∞, τ) = 0,

Π(ξ, 0) =

{

−1 ξ < ln((1 + rT )/(1 + qT )),0 ξ > ln((1 + rT )/(1 + qT )),

(87)

(0) = (1 + rT )/(1 + qT ),

where

a(ξ, τ) =ρ(τ)

ρ(τ)+ r − q − σ2

2− ρe−ξ − 1

T − τ.

5.4 An approximation scheme for pricing American style of Asian options

Similarly as in the case of a nonlinear Black–Scholes equation (see Section 4) we restrictthe spatial domainξ ∈ (0,∞) to a finite interval of valuesξ ∈ (0, L) whereL > 0 issufficiently large. Letk > 0 denote by the time step,k = T/m, and, byh > 0 the spatialstep,h = L/n wherem,n ∈ N again stand for the number of time and space discretizationsteps, resp. We denote byΠj

i an approximation ofΠ(ξi, τj), j ≈ (τj) whereξi = ih andτj = jk. Then for the Euler backward in time finite difference approximation of equation(60) we have

Πj − Πj−1

k+ bj ∂Πj

∂ξ−(

σ2

2+

ρje−ξ − 1

T − τj

)

∂Πj

∂ξ− σ2

2

∂2Πj

∂2ξ+

(

r +1

T − τj

)

Πj = 0

(88)wherebj is an approximation of the valueb(τj) where the functionb(τ) is defined as in

(56), i.e. b(τ) = ˙(τ)(τ) + r − q. The solutionΠj = Πj(x) is subject to Dirichlet boundary

conditions atξ = 0 andξ = L. We setΠ0(ξ) = Π(ξ, 0) (see (87). Again we split the aboveproblem into a convection part and a diffusive part by introducing an auxiliary intermediatestepΠj− 1

2 :(Convective part)

Πj− 12 − Πj−1

k+ bj∂xΠj− 1

2 = 0 , (89)

(Diffusive part)

Πj − Πj− 12

k−(

σ2

2+

ρje−ξ − 1

T − τj

)

∂Πj

∂ξ− σ2

2

∂2Πj

∂2ξ+

(

r +1

T − τj

)

Πj = 0. (90)

The convective part can be approximated by an explicit solution to the transport equation∂τ Π + b(τ)∂ξΠ = 0 for ξ > 0 and τ ∈ (τj−1, τj ] subject to the boundary condition

42 DanielSevcovic

Π(0, τ) = −1 and the initial conditionΠ(ξ, τj−1) = Πj−1(ξ). In contrast to classicalplain vanilla call or put options the free boundary function(τ) need not be monotonicallyincreasing (see e.g. [15] or [28]). Therefore depending on whether the termb(τ) = ˙(τ)

(τ) +

r − q is positive or negative the boundary conditionΠ(0, τ) = −1 at ξ = 0 is either in-flowing (b > 0) or out-flowing (b < 0). It means that the boundary conditionΠ(0, τ) = −1can be prescribed only ifb(τ) ≥ 0. Let us denote byB(τ) the primitive function tob(τ),i.e. B(τ) = ln (τ) + (r − q)τ . Solving the equation∂τ Π + b(τ)∂ξΠ = 0 we obtain:Π(ξ, τ) = Πj−1(ξ − B(τ) + B(τj−1)) if ξ − B(τ) + B(τj−1) > 0 andΠ(ξ, τ) = −1

otherwise. Hence the full time-space approximation of the half-step solutionΠj− 1

2i can be

obtained from the formula

Πj− 1

2i =

{

Πj−1(ηi) if ηi = ξi − ln j + ln j−1 − (r − q)k > 0 ,−1 otherwise.

(91)

In order to compute the valueΠj−1(ηi) we make use of a linear approximation betweendiscrete valuesΠj−1

i , i = 0, 1, ..., n.Using central finite differences for approximation of the derivative ∂xΠj we can ap-

proximate the diffusive part of a solution (90) as follows:

Πji − Π

j− 12

i

k+

(

r +1

T − τj

)

Πji (92)

−(

σ2

2+

ρje−ξi − 1

T − τj

)

Πji+1 − Πj

i−1

2h− σ2

2

Πji+1 − 2Πj

i + Πji−1

h2= 0 .

Therefore the vector of discrete valuesΠj = {Πji , i = 1, 2, ..., n} at the time levelj ∈

{1, 2, ...,m} is a solution to a tridiagonal system of equations

αjiΠ

ji−1 + βj

i Πji + γj

i Πji+1 = Π

j− 12

i (93)

for i = 1, 2, ..., n, where

αji ≡ αj

i (j) = − k

2h2σ2 +

k

2h

(

σ2

2+

ρje−ξi − 1

T − τj

)

,

γji ≡ γj

i (j) = − k

2h2σ2 − k

2h

(

σ2

2+

ρje−ξi − 1

T − τj

)

, (94)

βji ≡ βj

i (j) = 1 +

(

r +1

T − τj

)

k − (αji + γj

i ) .

The initial and boundary conditions atτ = 0 andx = 0, L, can be approximated as follows:

Π0i =

{

−1 for ξi < ln ((1 + rT )/(1 + qT )) ,0 for ξi ≥ ln ((1 + rT )/(1 + qT )) ,


for i = 0, 1, ..., n, andΠj0 = −E, Πj

n = 0. The equation for the free boundary position can be approximated by means of a finite difference approximation of ∂xΠ at the originξ = 0 as follows:

j =1 + r(T − τj) + (T − τj)

σ2

2Πj

1−Πj0

h

1 + q(T − τj). (95)

We formally rewrite discrete equations (91), (93) and (95) in the operator form:

j = F(Πj),

Πj− 12 = T (Πj , j), (96)

A(j)Πj = Πj− 12 ,

whereF(Πj) is the right-hand side of the algebraic equation (95),T (Πj , j) is the transportequation solver given by the right-hand side of (91) andA = A(j) is a tridiagonal matrixwith coefficients given by (94). The system (96) can be approximately solved by means ofsuccessive iterates procedure. We define, forj ≥ 1, Πj,0 = Πj−1, j,0 = j−1. Then the(p + 1)-th approximation ofΠj andj is obtained as a solution to the system:

j,p+1 = F(Πj,p),

Πj− 12,p+1 = T (Πj,p)j,p+1), (97)

A(j,p+1)Πj,p+1 = Πj− 12,p+1 .

Now, if the sequence of approximate discretized solutions{(Πj,p, j,p)}∞p=1 converges tosome limiting value(Πj,∞, j,∞) asp → ∞ then this limit is a solution to a nonlinear sys-tem of equations (96) at the time levelj and we can proceed by computing the approximatesolution in the next time levelj + 1.

5.5 Computational examples of the free boundary approximation

We end this section with several computational examples documenting the capability of thenew method for valuing early exercise boundary for Americanstyle of Asian call optionswith arithmetically averaged floating strike.

In Fig. 14 we show the behavior of the early exercise boundaryfunction (τ) and thefunction xf (t) = (T − t). In these numerical experiments we choser = 0.06, q =0.04, σ = 0.2 and very long expiration timeT = 50 years. These parameters correspondto the example presented in the preprint version of the paper[15] by Dai and Kwok. Asfar as other numerical parameters are concerned, we chose the mesh ofn = 100 spatialgrid points and we considered the number of time stepsm = 2 × 106 in other to achievevery fine time stepping corresponding to 13 minutes between consecutive time steps whenexpressed in the original time scale of the problem. In orderto make a graphical comparisonof the state variablex = S/A to its reciprocal valueA/S considered in [15] we also plotthe function1/xf (t).

44 DanielSevcovic

Figure 14:The function(τ) (above) and the free boundary positionxf (t) = (T − t) (bottom-right) and the plot of the function1/xf(t) = 1/(T − t) (bottom-left).

Figure 15:A 3D plot (left) and contour plot (right) of the functionΠ(ξ, τ).


Figure 16:Profiles of the functionΠ(ξ, τ) for various timesτ ∈ [0, T ].

Figure 17:A comparison of the free boundary positionxf (t) = (T − t) (left) and1/xf (t) =1/(T − t) (right) obtained by our method (solid curve) and that of the projected successive overrelaxation algorithm from [15] (dashed curve).

In Fig. 15 we can see the behavior of the transformed functionΠ in both 3D as well ascontour plot perspectives. Fig. 16 depicts the initial condition Π(ξ, 0) and five time stepsof the functionξ 7→ Π(ξ, τj) for τj = 0.1, 1, 5, 25, 50. A comparison of the free boundarypositionxf (t) = (T − t) as well as of its reciprocal value1/xf (t) = 1/(T − t) obtainedby our method (solid curve) and that of the projected successive over relaxation algorithmfrom [15] (dashed curve) is shown in Fig. 17. It is clear that our method and that of [15] givealmost the same result in the one third of the time interval[0, T ] close to the expiration timeT = 50. On the other hand, the long time behavior whenτ → T is quite different. Noticethat limτ→T (τ) can be analytically computed and is equal to1 (see [28] or [48]). Hencethe long time behavior of seems to be better approximated by our method. A comparisonof early exercise profiles with respect to varying dividend rateq is shown in Fig. 18.

Finally, we present numerical experiments for shorter expiration timesT = 0.5833(seven months) andT = 1 (one year) with zero dividend rateq = 0 andr = 0.05, σ = 0.2.

46 DanielSevcovic

Figure 18:A comparison of the free boundary positionxf (t) = (T − t) for various dividend yieldratesq = 0.04, 0.035, 0.3, 0.25.

Figure 19: The free boundary positionxf (t) = (T − t) for Asian call on assets paying nodividends (q = 0) with expiration timeT = 0.7 (left) andT = 1 (right).


Conclusion

In this survey paper we presented recent developments in fixed domain transformationmethods applied to evaluation of the early exercise boundary for American style of op-tions. We discussed an iterative numerical scheme for approximating of the early exerciseboundary for a class of Black–Scholes equations with a volatility which may depended onthe asset price as well as the second derivative of the optionprice. The method consisted oftransformation the free boundary problem for the early exercise boundary position into a so-lution of a nonlinear parabolic equation and a nonlinear algebraic constraint equation. Thetransformed problem has been solved by means of operator splitting iterative technique. Wealso presented results of numerical approximation of the free boundary for several nonlin-ear Black–Scholes equation including, in particular, Barles and Soner model and the Riskadjusted pricing methodology model. The method of fixed domain transformation has beenalso applied for evaluation of early exercise boundary for American style of Asian optionwith arithmetically averaged strike price.

Acknowledgments

The author thanks Matthias Ehrhardt for fruitful discussions and his encouragement to writethis survey chapter. I also appreciate help of my student B. Kucharcık with preparation ofthe last section. The work was supported by grants VEGA 1/3767/06 and DAAD-MSSR-11/2006.

References

[1] G. Alobaidi, R. Mallier and S. Deakin,Laplace transforms and installment options,Math. Models & Methods in Appl. Science18(8), (2004) 1167–1189.

[2] J. Ankudinova and M. Ehrhardt,On the numerical solution of nonlinear Black-Scholesequations, to appear in: Computers and Mathematics with Applications(2008).

[3] J. Ankudinova and M. Ehrhardt,Fixed domain transformations and highly accuratecompact schemes for Nonlinear Black-Scholes equations forAmerican Options, in M.Ehrhardt (ed.), Nonlinear Models in Mathematical Finance:New Research Trends inOption Pricing, Nova Science Publishers, Inc., Hauppauge,to appear fall 2008.

[4] M. Avellaneda and A. Paras,Dynamic Hedging Portfolios for Derivative Securitiesin the Presence of Large Transaction Costs, Applied Mathematical Finance,1 (1994)165–193.

[5] M. Avellaneda, A. Levy and A. Paras,Pricing and hedging derivative securities inmarkets with uncertain volatilities, Applied Mathematical Finance2 (1995) 73–88.

48 DanielSevcovic

[6] B. Barone-Adesi and R.E. Whaley,Efficient analytic approximations of American op-tion values, J. Finance42 (1987) 301–320.

[7] F. Black and M. Scholes,The pricing of options and corporate liabilities, J. PoliticalEconomy81 (1973) 637–654.

[8] G. Barles and H.M. Soner,Option Pricing with transaction costs and a nonlinearBlack–Scholes equation, Finance Stochast.,2 (1998) 369-397.

[9] L.A. Bordag and A.Y. Chmakova,Explicit solutions for a nonlinear model of financialderivatives, International Journal of Theoretical and Applied finance10 (2007) 1–21.

[10] L.A. Bordag and R. Frey,Nonlinear option pricing models for illiquid markets: scal-ing properties and explicit solutions, arxiv.org/abs/0708.1568 (2007)

[11] P. Carr, R. Jarrow and R. Myneni,Alternative characterizations of American put op-tions, Mathematical Finance2 (1992) 87–105

[12] J. Chadam,Free Boundary Problems in Mathematical Finance, Progress in IndustrialMathematics at ECMI 2006, Vol 12, Springer Berlin Heidelberg, 2008.

[13] Xinfu Chen, J. Chadam, Lishang Jiang and Weian Zheng,Convexity of the ExerciseBoundary of the American Put Option on a Zero Dividend Asset, Mathematical Fi-nance2008(2008) 185–197.

[14] C.K. Cho, S. Kang, T. Kim and Y. Kwon,Parameter estimation approach to the freeboundary for the pricing of an American call option, Computers and Mathematicswith applications51 (2006) 713–720.

[15] M. Dai, Y.K. Kwok, Characterization of optimal stopping regions of American Asianand lookback options, Mathematical Finance16 (2006) 63–82.

[16] J.N. Dewynne, S.D. Howison, J. Rupf and P. Wilmott,Some mathematical results inthe pricing of American options, Euro. J. Appl. Math.4 (1993) 381–398.

[17] B. During, M. Fournier and A. Jungel,High order compact finite difference schemesfor a nonlinear Black–Scholes equation, Int. J. Appl. Theor. Finance7 (2003) 767–789.

[18] M. Ehrhardt and P. Mickens,A fast, stable and accurate numerical method for theBlack-Scholes equation of American options, to appear in: International Journal ofTheoretical and Applied Finance.

[19] E. Ekstrom and J. Tysk,The American put is log-concave in the log-price, Journal ofMathematical Analysis and Appl.314(2) (2006) 710–723.

[20] E. Ekstrom,Convexity of the optimal stopping boundary for the Americanput option,Journal of Mathematical Analysis and Appl.299(1) (2004) 147–156.


[21] J.D. Evans, R. Kuske and J.B. Keller,American options on assets with dividends nearexpiry, Mathematical Finance12 (2002) 219–237.

[22] R. Frey and P. Patie,Risk Management for Derivatives in Illiquid Markets: A Simula-tion Study, Advances in Finance and Stochastics, Springer, Berlin, (2002) 137–159.

[23] R. Frey, and A. StremmeMarket Volatility and Feedback Effects from Dynamic Hedg-ing, Mathematical Finance4 (1997) 351–374.

[24] R. Geske and H.E. Johnson,The American put option valued analytically, J. Finance39 (1984) 1511–1524.

[25] R. Geske and R. Roll,On valuing American call options with the Black–Scholes Eu-ropean formula, J. Finance89 (1984) 443–455.

[26] P. Grandits and W. Schachinger,Leland’s approach to option pricing: the evolution ofa discontinuity, Mathematical Finance11(3) (2001) 347–355.

[27] S. Gripenberg S.O. London and O. Staffans,Volterra Integral and Functional Equa-tions, Cambridge University Press, 1990.

[28] A.T. Hansen and P.L. Jorgensen,Analytical Valuation of American-Style Asian Op-tionsl, Management Science46 (2000) 1116–1136.

[29] T. Hoggard, A.E. Whalley and P. Wilmott,Hedging option portfolios in the presenceof transaction costs, Advances in Futures and Options Research7 (1994) 21–35.

[30] J. Hull,Options, Futures and Other Derivative Securities, third edition, Prentice-Hall,1997.

[31] H. Imai, N. Ishimura, H. Sagakuchi,Computational technique for treating the non-linear Black-Scholes equation with the effect of transactional costs, Kybernetika432007 807–816.

[32] M. Jandacka and D.Sevcovic,On the risk adjusted pricing methodology based val-uation of vanilla options and explanation of the volatilitysmile, Journal of AppliedMathematics3 (2005) 235–258.

[33] H. Johnson,An analytic approximation of the American put price, J. Finan. Quant.Anal. 18 (1983) 141–148.

[34] C. Knessl,A note on a moving boundary problem arising in the American put option,Studies in Applied Mathematics(107) (2001) 157–183.

[35] I. Karatzas,On the pricing American options, Appl. Math. Optim.17 (1988) 37–60.

[36] R.A. Kuske and J.B. Keller,Optimal exercise boundary for an American put option,Applied Mathematical Finance5 (1998) 107–116.

50 DanielSevcovic

[37] M. Kratka,No Mystery Behind the Smile, Risk9 (1998) 67–71.

[38] Y.K Kwok, Mathematical Models of Financial Derivatives, Springer-Verlag, 1998.

[39] H.E. Leland,Option pricing and replication with transaction costs, Journal of Finance40 (1985) 1283–1301.

[40] L.W. MacMillan, Analytic approximation for the American put option, Adv. in FuturesOptions Res.1 (1986) 119–134.

[41] R. Mallier, Evaluating approximations for the American put option, Journal of Ap-plied Mathematics2 (2002) 71–92.

[42] R. Mallier and G. Alobaidi,The American put option close to expiry, Acta Mathemat-ica Univ. Comenianae73 (2004) 161–174.

[43] G.N. Milstein, O. Reiss and J. Schoenmakers,A new Monte Carlo method for Ameri-can Options, International Journal of Theoretical and Applied Finance7 (2004) 591–614.

[44] R. Mynemi,The pricing of the American option, Annal. Appl. Probab.2 (1992) 1–23.

[45] R. Roll,An analytic valuation formula for unprotected American call options on stockwith known dividends, J. Finan. Economy5 (1977) 251–258.

[46] D. Sevcovic,Analysis of the free boundary for the pricing of an American call option,Euro. Journal on Applied Mathematics12 (2001) 25–37.

[47] D. Sevcovic,An iterative algorithm for evaluating approximations to the optimal exer-cise boundary for a nonlinear Black-Scholes equation, Canad. Appl. Math. Quarterly15 (2007) 77–97.

[48] D. Sevcovic,Early exercise boundary for Asian options, in preparation.

[49] P. Schonbucher and P. Wilmott,The feedback-effect of hedging in illiquid markets,SIAM Journal of Applied Mathematics61 (2000) 232–272.

[50] R. Stamicar, D.Sevcovic and J. Chadam,The early exercise boundary for the Ameri-can put near expiry: numerical approximation, Canad. Appl. Math. Quarterly7 (1999)427–444.

[51] E.M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces,Princeton University Press, 1971.

[52] M. Widdicks, P.W. Duck, A.D. Andricopoulos and D.P. Newton, The Black-Scholesequation revisited: Asyptotic expansions and singular perturbations, MathematicalFinance15 (2005) 373–391.


[53] P. Wilmott, J. Dewynne and S.D. Howison,Option Pricing: Mathematical Modelsand Computation, UK: Oxford Financial Press, 1995.

[54] Jichao Zhao, R.M. Corless and M. Davison,Compact finite difference method forAmerican option pricing, Journal of Computational and Applied Mathematics206(2007) 306–321.

[55] S.P. Zhu,A new analytical approximation formula for the optimal exercise boundaryof American put options, International Journal of Theoretical and Applied Finance9(2006) 1141–1177.

[56] S.P. Zhu,Calculating the early exercise boundary of American put options with anapproximation formula, International Journal of Theoretical and Applied Finance10(2007) 1203–1227.

c 2 ˘3/ 4 @ 4 . 8 d b =/ ˘ ˜?5˜#1/ - uniba.sk · been generalized for more complex option...

Documents